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annotate texts/archimedesOldCVSRepository/archimedes/raw/monte_mecha_01_la_1577.raw @ 11:35edd67cabf9
Appendix Version 2009-02-14
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
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| date | Thu, 02 May 2013 11:12:52 +0200 |
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1 <pb id="p.0001"> |
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2 <head>GVIDIV BALDI |
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3 E MARCHIONIBVS |
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4 MONTIS |
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5 MECHANICORVM |
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6 LIBER.</head> |
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7 <fig> |
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8 <head>PISAVRI |
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9 Apud Hieronymum Concordiam.</head> |
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10 <head>M. D. LXXVII.</head> |
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11 <head>Cum Licentia Superiorum.</head> |
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12 <pb> |
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13 <fig> |
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14 <pb> |
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15 <head>GVIDIV BALDI |
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16 E MARCHIONIBVS |
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17 MONTIS |
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18 MECHANICORVM |
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19 LIBER.</head> |
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20 <fig> |
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21 <head>PISAVRI |
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22 Apud Hieronymum Concordiam.</head> |
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23 <head>M. D. LXXVII.</head> |
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24 <head>Cum Licentia Superiorum.</head> |
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25 <pb> |
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26 <head>PRAESENTI OPERE |
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27 CONTENTA.</head> |
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28 <p>De Libra. |
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29 <p>De Vecte. |
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30 <p>De Trochlea. |
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31 <p>De Axe in peritrochio. |
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32 <p>De Cuneo. |
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33 <p>De Cochlea. |
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34 <pb> |
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35 <head>AD FRANCISCVM |
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36 MARIAM II |
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37 VRBINATVM |
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38 AMPLISSIMVM DVCEM |
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39 GVIDIVBALDI |
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40 E MARCHIONIBVS |
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41 MONTIS</head> |
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42 <head>PRAEFATIO.</head> |
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43 <p>DV AE res (AMPLISSIME PRIN- |
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44 CEPS) quæ ad conciliandas homi |
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45 nibus facultates, vtilitas nempè, & |
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46 nobilitas, plurimùm valere con$ue |
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47 uerunt. illæ ad exornandam mecha |
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48 nicam facultatem, & eam præ om- |
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49 nibus alijs appetibilern reddendam con$pira$$e |
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50 mihi videnturinam $i nobilitatem (quod pleriq; |
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51 modò faciunt) ortuip$o metimur, occurret hinc |
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52 Geometria, illinc verò Phi$ica; quorumgemina |
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53 to complexunobili$$ima artium prodit mechani- |
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54 ca. $i enim nobilitatem magis, tùm $tratæ materiæ, |
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55 tùm argumentorum nece$$itati (quod Ari$tote- |
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56 les fatetur aliquandò) relatam volumus, omnium |
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57 proculdubiò nobili$$imam per$piciemus. quæ |
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58 <foot><12> 2</foot> |
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59 <pb> |
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60 quidem non $olum geometriam (vt Pappus te$ta |
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61 tur) ab$oluit, & perficit; verùm etiam & phi$ica- |
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62 rum rerum imperium habet: quandoquidem |
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63 quodcunq; Fabris, Architectis, Baiulis, Agricolis, |
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64 Nautis, & quàm plurimis alijs (repugnantibus na- |
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65 turæ legibus) opitulatur; id omne mechanicum |
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66 e$t imperium. quippè quod aduer$us naturam |
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67 vel eiu$dem emulata leges exercet; $umma id |
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68 certè admiratione dignum; veri$$imum tamen, |
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69 & à quocunque liberaliter admi$$um, qui pri- |
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70 us ab Ari$totele didicerit, omnia mechanica, |
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71 tùm problemata, tùm theoremata ad rotundam |
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72 machinam reduci, atq; ideo illo niti principio, |
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73 nõ minus $en$ui, quàm rationi noto. Rotunda ma |
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74 china e$t mouenti$$ima, & quò maior, eò mouen- |
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75 tior. Verùm huic nobilitati adnexa e$t $umma re |
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76 rum ad vitam pertinentium vtilitas, quæ propte- |
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77 rea omnes alias à diuer$is artibus propagatas an- |
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78 tecellit; quòd aliæ facultates po$t mundi gene$im |
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79 longa temporis intercapedine $uos explicarunt |
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80 v$us; i$ta verò & in ip$is mundi primordijs ita fuit |
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81 hominibus nece$$aria, vt ea $ublata Sol de mun- |
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82 do $ublatus videretur. nam quacunq; nece$$ita- |
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83 te Adæ vita degeretur; & quamuis etiam ca$is |
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84 contectis $tramine, & angu$tis tugurijs, ac gurgu- |
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85 $tijs cœli de fenderet iniurias; $ic & in corporis ve |
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86 $titu, licetip$e nihil aliud $pectaret, ni$i vt imbres, |
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87 <pb> |
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88 vt niues, vt ventos; vt Solem, vtfrigus arceret; |
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89 quodcunque tamen id fuit, omne mechanicum |
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90 fuit. neq; tamen huic facultati contingit, quod |
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91 ventis $olet, qui cùm vndè oriuntur, ibi vehe- |
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92 menti$$imi $int, ad longinqua tamen fracti, de- |
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93 bilitatiquè perueniunt: $ed quod magnis flumini- |
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94 bus crebriu$ accidit, quæ cùm in ip$o ortu parua |
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95 $int, perpetuò tamen aucta, eò ampliori ferun |
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96 tur alueo, quò à fontibus $uis longius rece$$e- |
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97 runt. Nam & temporis progre$$u mechanica fa |
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98 cultas $ub iugo æquum arationis laborem di- |
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99 $pen$are, atque aratrum agris circumagere cæ- |
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100 pit. deinceps bigis, & quadrigis docuit comea |
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101 tus, merces, onera quælibet vehere, è finibus |
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102 no$tri$ ad finitimos populos exportare, & ex il |
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103 lis contra importare ad nos. præterea cùm iam |
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104 res non tantùm nece$$itate, verùm etiam orna- |
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105 tu, & commoditate metirentur, mechanicæ |
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106 fuit $ubtilitatis, quòd nauigia remo impellere- |
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107 mus; quòd gubernaculo exiguo in extrema pup |
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108 pi collocato ingentes triremium moles inflecte- |
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109 remus; quòd vnius $æpè manu pro multis fabro- |
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110 rum manibus modò pondera lapidum, & tra- |
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111 bium Fabris, & Architectis $ubleuaremus; mo- |
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112 dò tollenonis $pecie aquas è puteis olitoribus e- |
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113 xhauriremus. hinc etiam è liquidorum prælis vi |
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114 na, olea, vnguenta expre$$a, & quicquid liquo- |
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115 <pb> |
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116 ris habent, per$oluere domino compul$a. hinc |
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117 magnas arborũ, & marmorum moles duobus in |
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118 contrarias partes di$trah&etilde;tibus vectibus diremp- |
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119 $imus; hinc militiæ in aggeribus extruendis, in |
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120 con$erenda manu, in opugnando, propugnan- |
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121 doq; loca infinitæ ferè redundarunt vtilitates; |
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122 hinc demum Lignatores, Lapicidæ, Marmorarij |
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123 Vinitores, Olearij, Vnguentarij, Ferrarij, Auri |
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124 fices, Metallici, Chirurgi, Ton$ores, Pi$tores, Sar |
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125 tores, omnes deniq; opifices beneficiarij, tot, tan |
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126 taq; vitæ humanæ $uppeditarunt commoda. Eant |
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127 nunc noui logodedali quidam mechanicorum |
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128 contemptores, perfricent frontem, $i quam ha- |
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129 bent, & ignobilitatem, atquè inutilitatem fal$ò |
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130 criminari de$inant: quòd $i & adhuc id minimè |
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131 velint, eos quæ$o in in$citia $ua relinquamus: |
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132 Ari$totelemquè potius philo$ophorum cory- |
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133 phæum imitemur, cuius mechanici amoris ardo |
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134 rem acuti$$imæ illæ mechanicæ quæ$tiones po$te |
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135 ris traditæ $atis declarant: qua quidem laude |
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136 Platonem magnificè $uperauit; qui (vt te$tatur |
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137 Plutarcus) Architam, & Eudoxum mechanicæ |
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138 vtilitatem impen$ius colentes ab in$tituto deter |
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139 ruit; quòd nobili$$imam philo$ophorum po$$e$- |
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140 $ionem in vulgus indicarent, ac publicarent; & |
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141 velut arcana philo$ophiæ my$teria proderent. |
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142 res $anè meo quidem iudicio pro$us vituperan- |
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143 <pb> |
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144 da, ni$i fortè velimus tam nobilis di$ciplinæ con |
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145 templationem quidem ocio$am laudare; fructum |
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146 verò, & v$um, arti$q; finem improbare. $ed præ |
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147 omnibus mathematicis vnus Archimedes ore |
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148 laudandus e$t pleniore, quem voluit Deus in me- |
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149 chanicis velut ideam $ingularem e$$e, quam om- |
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150 nes earum $tudio$i ad imitandum $ibi propone- |
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151 rent. is enim Cœle$tem globum exiguo admo- |
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152 dum, fragili què vitreo orbe conclu$um ita efin- |
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153 xit, $imulatis a$tris viuum naturæ opus, ac iura |
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154 poli motibus certis adeò præ$eferentibus; vt |
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155 æmula naturæ manus tale de $e encomium $it |
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156 promerita: $ic manus naturam, vt natura ma- |
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157 num ip$a immitata putetur. is poli$pa$tu manu |
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158 leua, & $ola, quinquies millenum modiorum |
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159 pondus attraxit. nauem in $iccum litus eductam, |
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160 ac grauius oneratam $olus machinis $uis ad $e |
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161 perindè pertraxit, ac $i in mari remis, veli$uè |
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162 impul$a moueretur, quã & po$tea in litore (quod |
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163 omnes Siciliæ vires non potuerunt) in mare de- |
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164 duxit. ab i$to etiam ea extiterunt bellica tor- |
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165 menta, quibus Syracu$æ aduer$us Marcellum |
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166 ita defen$æ $unt, vt pa$$im eorum machinator |
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167 Briareus, & centimanus à Romanis appellare- |
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168 tur. demum hac arte confi$us eò proce$$it au- |
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169 daciæ, vt eam vocem naturæ legibus adeò re- |
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170 pugnantem protulerit. Da mihi, vbi $i$tam, ter |
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171 <pb> |
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172 ramq; mouebo. quod tamen non modò nos |
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173 vecte tantùm fieri potui$$e in præ$enti libro doce |
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174 mus; verùm etiam, & omnis anti quitas (quod |
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175 multis forta$$è mirabile videbitur) id penitus |
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176 credidi$$e mihi videtur; quæ Neptuno tri- |
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177 dentem tanquam vectem attribuit; cuius ope |
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178 terræ concu$$or vbiq; nuncupatur à poetis. ad |
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179 quod etiam a$piciens celeberrimus no$ter poeta |
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180 Neptunum inducit i$ta machina $yrtes, quò ma- |
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181 gis apparerent Troianis, $ubleuantem. |
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182 <p>“Leuat ip$e tridenti |
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183 & va$tas aperit $yrtes.” |
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184 <p>Mechanici præterea fuerunt Heron, Cte$ibius, |
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185 & Pappus, qui licet ad mechanicæ apicem, perin- |
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186 de atq; Archimedes, euecti forta$$è minimè $int; |
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187 mechanicam tamen facultatem egregiè percal- |
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188 luerunt; tale$q; fuerunt, & præ$ertim Pappus, vt |
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189 eum me ducem $equentem nemo (vt opinor) cul |
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190 pauerit. quod & propterea libentius feci, quòd |
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191 nè latum quidem vnguem ab Archimedeis prin- |
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192 cipijs Pappus recedat. ego enim in hac præ$ertim |
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193 facultate Archimedis ve$tigijs hærere $emper vo |
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194 lui: & licet eius lucubrationes ad mechanicã per- |
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195 <pb> |
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196 tinentes multis ab hinc annis pa$$im $oleant do- |
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197 ctis de$iderari: eruditi$$imus tamen libellus de æ- |
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198 queponderantibus præ manibus hominũ adhuc |
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199 ver$atur, in quò tanquam in copio$i$$ima pœnu |
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200 omnia ferè mechanica dogmata repo$ita mihi vi- |
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201 dentur; quem $anè libellum, $i ætatis no$træ mathe |
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202 matici $ibi magis familiarem adhibui$$ent; reperi$ |
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203 $ent $anè $ent&etilde;tias multas, quas modó ip$i firmas, |
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204 & ratas e$$e docent; $ubtili$$imè, atquè veri$- |
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205 $imè conuul$as, & labefactatas. $ed hoc vi- |
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206 derint ip$i. ego enim ad Pappum redeo, qui |
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207 ad v$um mathematicarum vberiorem, emulu- |
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208 mentorumquè acce$$iones amplificandas peni- |
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209 tus conuer$us, de quinque principibus machi- |
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210 nis, Vecte nempè, Trochlea, Axe in peri- |
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211 trochio, Cuneo, & Cochlea, multa egre- |
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212 giè philo$ophatus e$t; demon$trauit què quicquid |
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213 in machinis, aut cogitari peritè, aut acutè |
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214 definiri, aut certò $tatui pote$t, idomne quin- |
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215 què illis infinita vi præditis machinis referen- |
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216 dum e$$e. atquè vtinam iniuria temporis ni- |
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217 hil è tanti viri $criptis abra$i$$et: nec enim tam |
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218 den$a in$citiæ caligo vniuer$um propè terra- |
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219 rum orbem obtexi$$et, neque tanta mechani |
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220 cæ facultatis e$$et ignoratio con$ecuta, vt ma- |
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221 thematicarum proceres exi$timarentur illi, qui |
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222 modò inepti$$ima quadam di$tinctione, diffi- |
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223 <foot><12><12></foot> |
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224 <pb> |
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225 cultates nonnullas, nec illas tamen $atis ar- |
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226 duas, & ob$curas è medio tollunt. reperiun- |
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227 tur enim aliqui, no$traq; ætate emunctæ naris |
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228 mathematici, qui mechanicam, tùm mathe- |
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229 maticè $eor$um, tùm phi$icè con$iderari po$- |
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230 $e affirmant; ac $i aliquando, vel $ine demon |
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231 $trationibus geometricis, vel $ine vero motu |
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232 res mechanicæ con$iderari po$$int: qua $anè di- |
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233 $tinctione (vt leuius cum illis agam) nihil aliud mi- |
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234 hi commini$ci videntur, quàm vt dum $e, tùm |
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235 phi$icos, tùm mathematicos proferant, vtra- |
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236 que (quod aiunt) $ella excludantur. nequè |
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237 enim amplius mechanica, $i à machinis ab$tra |
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238 hatur, & $eiungatur, mechanica pote$t appel |
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239 lari. Emicuit tamen inter i$tas tenebras (quam- |
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240 uis alij quoquè nonnulli fuerint præclari$$imi) |
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241 Solis in$tar Federicus Commandinus, qui multis |
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242 docti$$imis elucubrationibus ami$$um mathema |
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243 ticarum patrimonium non modò re$taurauit, |
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244 verùm etiam auctiùs, & locupletiùs effecit. |
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245 erat enim $ummus i$te vir omnibus adeò facul- |
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246 tatibus mathematicis ornatus, vt in eo Archi- |
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247 tas, Eudoxus, Heron, Euclides, Theon, Ari- |
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248 $tarcus, Diophantus, Theodo$ius, Ptolemæus |
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249 Apollonius, Serenus, Pappus, quin & ip- |
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250 $emet Archimedes ($iquidem ip$ius in Archi- |
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251 medem $cripta Archimedis olent lucernam) re |
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252 <pb> |
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253 uixi$$e viderentur. & ecce repentè è tenebris (vt |
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254 confidimus) ac vinculis corporis in lucem, li- |
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255 bertatem què productus mathematicas alieni$- |
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256 $imo tempore optimo, & præ$tanti$$imo patre |
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257 orbatas, nos verò ita con$ternatos reliquit, vt e- |
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258 ius de$iderium vix longo $ermone mitigare |
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259 po$$e videamur. Ille tamen perpetuò in alia- |
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260 rum mathematicarum explicationem ver$ans, |
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261 mechanicam facultatem, aut penitus præter- |
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262 mi$it, aut modicè attigit. Quapropter in hoc |
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263 $tudium ardentiùs ego incumbere cæpi, nec me |
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264 vnquam per omne mathematum genus vagan |
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265 tem ea $olicitudo de$eruit; ecquid ex vno |
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266 quoquè decerpi, ac delibari po$$it; quo ad me |
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267 chanicam expoliendam, & exornandam acco- |
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268 modatior e$$e po$$em. Nunc verò cùm mihi |
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269 videar, noni ea quidem omnia, quæ ad mecha |
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270 nicam pertinent, perfeci$$e; $ed eò v$q; tamen |
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271 progre$$us, vtijs, qui ex Pappo, ex Vitruuio, |
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272 & ex alijs didicerint, quid $it Vectis, quid Tro- |
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273 chlea, quid Axis in peritrochio, quid Cuneus, |
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274 quid Cochlea; quomodoq; vt pondera moueri |
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275 po$$int, aptari debeant; adhuc tamen acciden- |
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276 tia permulta, quæ inter potentiam, & pondus |
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277 vectis virtute illis in$unt in$trumentis, perdi$ce- |
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278 re cupiunt, opis aliquid adferre po$$im; putaui |
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279 tempus iam po$tulare, vt prodirem; & nauatæ |
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280 <foot><12><12> 2</foot> |
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281 <pb> |
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282 in hoc genere operæ $pecimen aliquod darem. |
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283 Verùm quò facilius totius operis $ub$tructio |
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284 ad fa$tigium $uum per duceretur, nonnulla quo- |
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285 què de libra fuerunt pertractanda, & præ$er- |
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286 tim dum vnico pondere alterum $olum ip$ius |
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287 brachium penitus deprimitur: que in re mi- |
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288 rum e$t quantas fecerint ruinas Iordanus (qui |
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289 inter recentiores maximæ fuit auctoritatis) & |
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290 alij; qui hanc rem $ibi di$cutiendam propo$ue |
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291 runt. opus $anè arduum, & for$an viribus no- |
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292 $tris impar aggre$si $umus; in eo tamen digni, vt |
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293 no$tros conatus, & indu$triam ad præclara ten |
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294 dentem bonorum omnium perpetuus applau- |
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295 $us, approbatioq; comitetur; quòd ad $tudium |
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296 tàm illu$tre, tam magnificum, tam laudabile |
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297 contulimus quicquid habuimus virium. quod |
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298 $anè qualecunq; $it, tibi celeberrime PRINCEPS |
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299 nuncupandum cen$uimus; cuius $anè con$ilij, |
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300 atq; in$tituti no$tri rationes multas reddere in |
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301 promptu e$t: & primùm hæreditaria tibi in fa- |
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302 miliam no$tram promerita, quibus nos ita de- |
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303 uictos habes; vt facilè intelligamus ad fortunas |
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304 non modò no$tras, verùm & ad $anguinem, & |
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305 vitam quoq; pro tua dignitate propendendam |
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306 parati$$imos e$$e debere. Præterea illud non |
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307 parui quoq; ponderis accedit, quòd à pueri- |
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308 tia literarum omnium, $ed præcipuè mathe- |
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309 <pb> |
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310 maticarum de$iderio ita fueris incen$us, vt ni- |
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311 $i illis adeptis vitam tibi acerbam, atq; in$ua- |
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312 uem $tatueres. proinde in earum $tudio infi- |
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313 xus primam ætatis partem in illis percipiendis |
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314 exegi$ti, eamquè $æpius verè principe dignam |
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315 vocem protuli$ti, te propterea mathematicis |
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316 præ$ertim delectari, quòd i$tæ maximè ex do- |
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317 me$tico illo, & vmbratili vitæ genere in Solem |
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318 (quod dicitur) & puluerem prodire po$sint: cu |
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319 ius $anè rei tuum flagranti$simum ab ineunte æta |
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320 te peritiæ militaris de$iderium, exploratum in- |
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321 dicium poterat e$$e, ni$i nimis emendicatæ men- |
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322 tis e$$et ea proponere, quæ à te $perari po$$ent; |
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323 quando tu penitus adole$cens, egregia multa fa |
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324 cinora proficere matura$ti. Tu enim cùm iam |
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325 à $ancti$$imo Pontifice Pio V $aluberrimæ Prin- |
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326 cipum Chri$tianorum coniunctionis fundamen- |
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327 ta iacta e$$ent, alacer admodum ad debellan- |
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328 dos Chri$ti ho$tes profectus, $olidi$$imam, ac ve- |
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329 ri$$imam gloriam tibi compara$ti. Tu quoties de |
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330 $umma rerum deliberatum e$t, eas $ententias |
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331 dixi$ti, quæ $ummam prudentiam cùm $umma |
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332 animi excel$itate coniunctam indicarent. ommit- |
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333 taminterim pleraq; alia illis temporibus egre- |
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334 giè, viriliter què à te ge$ta, ne tibi ip$iea, quæ |
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335 omnibus $unt manife$ta, palàm facere videar: |
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336 <pb> |
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337 quæ cùm omnia magna, & præclara $int; mul- |
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338 tò tamen à te maiora, & præclara expectant |
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339 adhuc homines. Vale interim præ$tanti$$imum |
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340 orbis decus, & $i quando aliquid otij nactus |
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341 fueris has meas vigiliolas a$picere ne dedi- |
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342 gneris. |
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343 <pb n=1> |
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344 <head>GVIDIVBALDI |
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345 E MARCHIONIBVS |
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346 MONTIS.</head> |
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347 <head>MECHANICORVM |
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348 LIBER.</head> |
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349 <fig> |
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350 <head>DEFINITIONES.</head> |
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351 <p>Centrvm grauitatis vniu$cu- |
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352 iu$q; corporis e$t punctum quod- |
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353 dam intra po$itum, à quo $i gra- |
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354 ue appen$um mente concipiatur, |
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355 dum fertur, quie$cit; & $eruat eam, |
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356 quam in principio habebat po$i- |
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357 tionem: neq; in ip$a latione circumuertitur. |
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358 <p>Hanc centri grauitatis definitionem Pappus Alexandrinus in |
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359 octauo Mathematicarum collectionum libro tradidit. Federicus |
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360 verò Commandinus in libro de centro grauitatis $olidorum idem |
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361 centrum de$cribendo ita explicauit. |
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362 <p>Centrum grauitatis vniu$cuiu$q; $olidæ figu- |
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363 ræ e$t punctum illud intra po$itum, circa quod |
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364 vndiq; partes æqualium momentorum con$i- |
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365 $tunt. $i enim per tale centrum ducatur planum |
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366 figuram quomodocunq; $ecans $emper in par- |
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367 tes æqueponderantes ip$am diuidet. |
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368 <foot>A</foot> |
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369 <pb> |
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370 <head>COMMVNES NOTIONES.</head> |
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371 <head>I</head> |
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372 <p>Si ab æqueponderantibus æqueponderantia au- |
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373 ferantur, reliqua æqueponderabunt. |
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374 <head>II</head> |
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375 <p>Si æqueponderantibus æqueponderantia adii- |
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376 ciantur, tota $imul æqueponderabunt. |
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377 <head>III</head> |
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378 <p>Quæ eidem æqueponderant, inter $e æquè $unt |
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379 grauia. |
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380 <head>SVPPOSITIONES.</head> |
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381 <head>I</head> |
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382 <p>Vnius corporis vnum tantùm e$t centrum gra- |
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383 uitatis. |
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384 <head>II</head> |
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385 <p>Vnius corporis centrum grauitatis $emper in |
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386 eodem e$t $itu re$pectu $ui corporis. |
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387 <head>III</head> |
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388 <p>Secundùm grauitatis centrum pondera deor- |
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389 $um feruntur. |
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390 <pb n=2> |
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391 <head>DE LIBRA.</head> |
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392 <p>Anteqvam de libra $ermo ha |
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393 beatur, vtres clarior eluce$cat, $it |
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394 libra AB recta linea; CD verò |
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395 trutina, quæ $ecundum commu- |
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396 nem con$uetudinem horizonti |
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397 $emper e$t perpendicularis. pun- |
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398 ctum autem C immobile, circa quod vertitur li- |
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399 bra, centrum libræ |
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400 vocetur. itidemque |
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401 (quamuis tamen im- |
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402 proprie) $iue $upra, |
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403 $iue infra libram fue |
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404 rit con$titutum. CA |
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405 verò, & CB, tum di |
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406 $tantiæ, tum libræ |
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407 brachia nuncupen- |
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408 tur. & $i à centro li- |
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409 bræ $upra, vel infra |
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410 <fig> |
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411 libram con$tituto ip$i AB perpendicularis duca- |
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412 tur, hæc perpendiculum vocetur, quæ libram AB |
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413 $ub$tinebit; & quocunque modo moueatur libra, |
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414 ip$i $emper perpendicularis exi$tet. |
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415 <foot>A 2</foot> |
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416 <pb> |
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417 <head>LEMMA.</head> |
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418 <p>Sit linea AB horizonti perpendicularis, & dia |
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419 metro AB circulus de$cribatur AEBD, cuius |
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420 centrum C. Dico punctum B infimum e$$e lo- |
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421 cum circumferentiæ circuli AEBD; punctum |
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422 verò A $ublimiorem; & quælibet puncta, vt DE |
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423 æqualiter à puncto A di$tantia æqualiter e$$e |
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424 deor$um; quæ verò propius $untip$i A eis, quæ |
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425 magis di$tant, $ublimiora e$$e. |
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426 <p>Producatur AB v$q; ad mundi cen- |
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427 trum, quod $it F; deinde in circuli circum- |
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428 <marg>8. <I>Tertil.</I></marg> ferentia quoduis accipiatur punctum G; |
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429 connectanturq; FG FD FE. Quoniam |
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430 n. BF minima e$t omnium, quæ à puncto |
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431 F ad circumferentiam AEBD ducun- |
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432 tur; erit BF ip$a FG minor. quare punctum |
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433 B propius erit puncto F, quàm G. hacq; |
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434 ratione o$tendetur punctum B quouis alio |
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435 puncto circumferentiæ circuli AEDB |
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436 mundi centro propius e$$e. erit igitur pun- |
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437 ctum B circumferentiæ circuli AEBD |
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438 infimus locus. Deinde quoniam AF per |
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439 centrum ducta maior e$t ip$a GF; erit |
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440 punctum A non $olũ ip$o G, verum etiam |
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441 quouis alio puncto circumferentiæ circuli |
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442 AEBD $ublimius. Præterea quoniam DF |
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443 FE $unt æquales; puncta DE æqualiter |
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444 <fig> |
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445 mundi centro di$tabunt. & cum DF maior $it FG; erit pun- |
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446 ctum D ip$i A propius puncto G $ublimius. quæ omnia demon- |
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447 $trare oportebat. |
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448 <pb n=3> |
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449 <head>PROPOSITIO I.</head> |
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450 <p>Si Pondus in eius centro grauitatis a recta $u- |
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451 $tineatur linea, nunquam manebit, ni$i eadem li- |
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452 nea horizonti fuerit per pendicularis. |
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453 <p>Sit pondus A, cuius centrum gra |
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454 uitatis B, quod à linea CE $u$ti- |
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455 neatur. Dico pondus nunquam |
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456 perman$urum, ni$i CB horizonti |
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457 perpendicularis exi$tat. $it pun- |
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458 ctum C immobile, quod vt pon |
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459 dus $u$tineatur, nece$$e e$t. & cum |
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460 punctum C $it immobile, $i pon- |
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461 dus A mouebitur, punctum B cir |
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462 culi circumferentiam de$cribet, |
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463 cuius $emidiameter erit CB. qua |
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464 re centro C, $patio verò BC, cir- |
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465 culus de$cribatur BFDE. $itq; |
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466 <fig> |
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467 primum BC horizonti perpendicularís, quæ víq; ad D produca- |
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468 tur; atq; punctum C $it infra punctum B. Quoniam enim pondus <marg><I>Supp.</I> 3. <I>huius.</I></marg> |
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469 A $ecundum grauitatis centrum B deor$um mouetur; punctum |
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470 B deor$um in centrum mundi, quò naturaliter tendit, per re- |
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471 ctam lineam BD mouebitur: totum ergo pondus A eius cen- |
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472 tro grauitatis B $uper rectam lineam BC graue$cet. cum au- |
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473 tem pondus à linea CB $u$tineatur, linea CB totum $u$ti- |
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474 nebit pondus A; $uper quam deor$um moueri non pote$t, cum |
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475 abip$a prohibeatur: per definitionem igitur centri grauitatis pun |
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476 ctum B, pondu$q; A in hoc $itu manebunt. & quamquam B quo- |
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477 cunq; alio puncto circuli $it $ublimius, ab hoc tamen $itu deor$um |
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478 per circuli circumferentiam nequaquam mouebitur non enim ver- |
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479 $us F magis, quàm ver$us E inclinabitur, cum ex vtraq; parte æqua- |
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480 lis $it de$cen$us; neq; pondus A in vnam magis, quàm in alteram |
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481 partem propen$ionem habeat: quod non accidit in quouis alio |
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482 puncto circumferentiæ circuli (præter D) $it ponderis eiu$dem |
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483 <pb> |
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484 centrum grauitatis, vt in F; cum ex |
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485 puncto F ver$us D $it de$cen$us, at |
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486 verò ver$us B a$cen$us. quare pun- |
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487 ctum F deor$um mouebitur. & quo |
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488 niam per rectam lineam in centrum |
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489 mundi moueri non pote$t, cum à |
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490 puncto C immobili propter lineam |
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491 CF prohibeatur; deor$um tamen |
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492 $icuti eius natura po$tulat, $emper |
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493 mouebitur. & cum infimus locus $it |
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494 D, per circumferentiã FD mouebi |
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495 tur, donec in D perueniat, in quo |
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496 $itu manebit, põdu$q; immobile exi |
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497 <fig> |
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498 $tet. tum quia deor$um amplius moueri non pote$t, cum ex pun- |
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499 cto C $it appen$um; tum etiam, quia in eius centro grauitatis $u$ti |
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500 netur. Quando autem F erit in D, erit quoq; linea FC in DC, |
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501 $imulq; horizonti perpendicularis. pondus ergo nunquam mane |
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502 bit, donec linea CF horizonti perpendicularis non exi$tat. quod |
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503 o$tendere oportebat. |
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504 <p>Ex hoc elici pote$t, pondus quocunq; modo |
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505 in dato puncto $u$tineatur, nunquam manere; ni |
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506 $i quando a centro grauitatis ponderis ad id pun |
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507 ctum ducta linea horizonti $it perpendicularis. |
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508 <p>Vt ii$dem po$itis, $u$tineatur |
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509 pondus à lineis CG CH. Dico |
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510 $i ducta BC horizonti $it perpen- |
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511 dicularis, pondus A manere. $i verò |
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512 ducta CF non $it horizonti per- |
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513 pendicularis, punctum F deor$um |
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514 v$q; ad D moueri; in quo $itu pon- |
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515 dus manebit, ductaq; CD horizon |
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516 ti perpendicularis exi$tet. quæ om- |
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517 nia eadem ratione o$tendentur. |
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518 <fig> |
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519 <pb n=4> |
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520 <head>PROPOSITIO II.</head> |
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521 <p>Libra horizonti æquidi$tans, cuius centrum |
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522 $it $upra libram, æqualia in extremitatibus, æqua |
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523 literq; à perpendiculo di$tantia habens pondera, |
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524 $i ab eiu$modi moueatur $itu, in eundem rur$us |
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525 relicta, redibit; ibíq; manebit. |
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526 <p>Sit libra AB recta li- |
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527 nea horizonti æquidi- |
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528 $tans, cuius centrum C |
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529 $it $upral ibram; $itq; CD |
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530 perpendiculũ, quod ho- |
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531 rizonti perpendiculare |
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532 erit: atq; di$tantia DA $it |
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533 di$tantiæ DB æqualis; |
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534 $intq; in AB pondera æ- |
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535 qualia, quorũ grauitatis |
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536 centra $int in AB pũctis. |
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537 Moueatur AB libra ab |
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538 <fig> |
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539 hoc $itu, putá in EF, deinde relinquatur. dico libram EF in AB ho |
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540 rizonti æquidi$tantem redire, ibíq; manere. Quoniam autem pun |
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541 ctum C e$t immobile, dum libra mouetur, punctum D circuli cir- |
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542 cumferentiam de$cribet, cuius $emidiameter erit CD. quare cen- |
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543 tro C, $patio verò CD, circulus de$cribatur DGH. Quoniam |
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544 enim CD ip$i libræ $emper e$t perpendicularis, dum libra erit in |
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545 EF, linea CD erit in CG, ita vt CG $it ip$i EF perpendicula- |
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546 ris. Cùm autem AB bifariam à puncto D diuidatur, & pondera |
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547 in AB $int æqualia; erit magnitudinis ex ip$is AB compo$itæ cen <marg>4. <I>primi Ar cbimedis de æqueponderantibus.</I></marg> |
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548 trum grauitatis in medio, hoc e$t in D. & quãdo libra vná cum pon |
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549 deribus erit in EF; erit magnitudinis ex vtri$q; EF compo$itæ cen |
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550 trum grauitatis G. & quoniam CG horizonti non e$t perpendi- <marg>1. <I>Huius</I></marg> |
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551 cularis; magnitudo ex ponderibus EF compo$ita in hoc $itu mi- |
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552 nimè per$i$tet, $ed deor$um $ecũdùm eius centrum grauitatis G per |
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553 circumferentiam GD mouebitur; donec CG horizonti fiat per- |
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554 <pb> |
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555 pendicularis, $cilicet do- |
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556 nec CG in CD redeat. |
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557 Quando autem CG erit |
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558 in CD, linea EF, cùm |
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559 ip$i CG $emper ad rectos |
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560 $it angulos, erit in AB; in |
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561 <marg>1. <I>Huius.</I></marg> quo $itu quoq; manebit. li |
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562 bra ergo EF in AB hori- |
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563 zonti æquidi$tãtem redi |
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564 bit, ibíq; manebit. quod |
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565 demon$trare oportebat. |
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566 <fig> |
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567 <head>PROPOSITIO III.</head> |
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568 <p>Libra horizonti æquidi$tans æqualia in extre- |
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569 mitatibus, æqualiterq; à perpendiculo di$tan- |
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570 tia habens pondera, centro infernè collocato, in |
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571 hoc $itu manebit. $i verò inde moueatur, deor- |
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572 $um relicta, $ecundùm partem decliuiorem mo- |
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573 uebitur. |
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574 <fig> |
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575 <p>Sit libra AB rectá li- |
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576 nea horizonti æquidi- |
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577 $tans, cuius centrum C |
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578 $it infra libram; perpen- |
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579 diculumq; $it CD, quod |
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580 horizonti perpendiculare |
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581 erit; & di$tantia AD $it |
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582 di$tantiæ DB æqualis; |
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583 $intq; in AB pondera |
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584 æqualia, quorum grauita- |
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585 tis centra $int in punctis |
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586 AB. Dico primùm libram AB in hoc $itu manere. Quoniam |
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587 enim AB bifariam diuiditur à puncto D, & pondera in AB $unt |
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588 æqualia; erit punctum D centrum grauitatis magnitudinis ex |
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589 <pb n=5> |
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590 vtri$q; AB ponderibus compo$itæ. & CD libram $u$tinens ho- <marg>4. <I>Primi Archim. de æquep.</I></marg> |
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591 rizonti e$t perpendicularis, libra ergo AB in hoc $itu manebit. <marg>1. <I>Huius.</I></marg> |
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592 moueatur autem libra AB ab hoc $itu, putà in EF, deinde relinqua |
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593 tur. dico libram EF ex parte F moueri. Quoniam igitur CD |
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594 ip$i libræ $emper e$t perpendicularis, dum libra erit in EF, erit |
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595 CD in CG ip$i EF perpendicularis. & punctum G magnitudi- |
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596 nis ex EF compo$itæ centrum grauitatis erit; quod dum moue- |
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597 tur, circuli circumferentiam de$cribet DGH, cuius $emidiameter |
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598 CD, & centrum C. Quoniam autem CG horizonti non e$t per- |
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599 pendicularis, magnitudo ex EF ponderibus compo$ita in hoc $i- |
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600 tu minimè manebit; $ed $ecundùm eius grauitatis centrum G deor |
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601 $um per circumferentiam GH mouebitur. libra ergo EF ex par |
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602 te F deor$um mouebitur, quod demon$trare oportebat. |
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603 <head>PROPOSITIO IIII.</head> |
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604 <p>Libra horizonti æquidi$tans æqualia in ex- |
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605 tremitatibus, æqualiterq; à centro in ip$a libra |
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606 collocato, di$tantia habens pondera; $iue inde |
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607 moueatur, $iue minus; vbicunq; reli cta, mane bit. |
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608 <fig> |
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609 <p>Sit libra recta linea A |
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610 B horizonti æquidi$tans, |
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611 cuius centrum C in ea- |
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612 dem $it linea AB; di$tan |
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613 tia verò CA $it di$tantiæ |
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614 CB æqualis: $intq; pon- |
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615 dera in AB æqualia, quo- |
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616 rum centra grauitatis $int |
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617 in puntis AB. Moueatur |
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618 libra, vt in DE, ibiquè |
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619 relinquatur. Dico primùm libram DE non moueri, in eoquè $itu |
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620 manere. Quoniam enim pondera AB $unt æqualia; erit magni- |
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621 tudinis ex vtroq; pondere, videlicet A, & B compo$itæ centrum |
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622 grauitatis C. quare idem punctum C, & centrum libræ, & centrũ |
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623 grauitatis totius ponderis erit. Quoniam autem centrum libræ |
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624 <foot>B</foot> |
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625 <pb> |
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626 C, dum libra AB vnà |
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627 cum ponderibus in DE |
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628 mouetur, immobile re- |
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629 manet, centrum quoq; |
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630 grauitatis, quod e$t idem |
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631 C, non mouebitur. nec |
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632 igitur libra DE mouebi |
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633 tur, per definitionem |
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634 centri grauitatis, cum in |
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635 ip$o $u$pendatur. Idip- |
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636 <fig> |
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637 $um quoq; contingit libra in AB horizonti æquidi$tante, vel in |
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638 quocunq; alio $itu exi$tente. Manebit ergolibra, vbi relinque- |
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639 tur. quod demon$trare oportebat. |
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640 <p>Cum verò in iis, quæ dicta $unt, grauitatis tantùm magnitudi |
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641 num, quæ in extremitatibus libræ po$itæ $unt æquales, ab$q; lí- |
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642 bræ grauitate con$iderauerimus; quoniam tamen adhuc libræ bra- |
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643 chia $unt æqualia, idcirco idem libræ, eius grauitate con$iderata, |
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644 vnà cum ponderibus, vel $ine ponderibus eueniet. idem enim cen |
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645 trum grauitatis fine ponderibus libræ tantùm grauitatis centrum |
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646 erit. Similiter $i pondera in libræ extremitatibus appendantur, vt |
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647 fieri $olet, idem cueniet; dummodo ex $u$pen$ionum punctis ad |
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648 centra grauitatum ponderum ductæ lineæ (quocunq; modo mo- |
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649 ueatur libra) $i protrahantur, in centrum mundi concurrant. vbi |
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650 enim pondera hoc modo $unt appen$a, ibi graue$cunt, ac$i in ii$- |
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651 dem punctis centra grauitatum haberent. præterea, quæ $equun- |
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652 tur, eodem pror$us modo con$iderare poterimus. |
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653 <p><marg><I>Iordanus de Ponderibus.</I></marg>Quoniam autem huic determinationi vltimæ multa à nonnullis |
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654 aliter $entientibus dicta officere videntur; idcirco in hac parte ali- |
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655 <marg><I>Hyerommus Carda nus de $ubtilitate.</I></marg> quantulum immorari oportebit; & pro viribus, non $olum pro- |
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656 priam $ententiam, $ed Archimedem ip$um, qui in hac eadem e$$e |
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657 <marg><I>Nicolaus Tartalea de quæ$itis, ac inuentio nibus.</I></marg> $ententia videtur, defendere conabor. |
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658 <pb n=6> |
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659 <fig> |
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660 <p>Ii$dem po$itis, duca- |
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661 tur FCG ip$i AB, & |
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662 horizonti perpendicula- |
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663 ris; & centro C, $patio- |
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664 què CA, circulus de$cri |
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665 batur ADFBEG. erunt |
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666 puncta ADBE in circu |
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667 li circumferentia; cum li- |
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668 bræ brachia $int æqualia. |
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669 & quoniam in vnam con |
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670 ueniunt $ententiam, a$$e- |
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671 rentes $cilicet libram DE |
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672 neq; in FG moueri, ne- |
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673 que in DE manere, $ed in AB horizonti æquidi$tantem rediré. |
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674 hanc corum $ententiam nullo modo con$i$tere po$$e o$tendam. |
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675 Non enim, $ed $i quod aiunt, euenerit, vel ideo erit, quia pondus |
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676 D pondere E grauius fuerit, vel $i pondera $unt æqualia, di$tantiæ, |
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677 quibus $unt po$ita, non erunt æquales, hoc e$t CD ip$i CE non erit |
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678 æqualis, $ed maior. Quòd autem pondera in DE $int æqualia, & |
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679 di$tantia CD $it æqualis di$tantiæ CE: hæc ex $uppo$itione pa- |
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680 tent. Sed quoniam dicunt pondus in D in eo $itu pondere in E |
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681 grauius e$$e in altero $itu deor$um: dum pondera $unt in DE, pun- |
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682 ctum C non erit amplius centrum grauitatis, nam non manent, $i |
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683 ex C $u$pendantur; $ed erit in linea CD, ex tertia primi Archi- |
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684 medis de æqueponderantibus. non autem erit in linea CE, cum pon |
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685 dus D grauius $it pondere E. $it igitur in H, in quo $i $u$pendan- |
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686 tur, manebunt. Quoniam autem centrum grauitatis ponderum |
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687 in AB connexorum e$t punctum C; ponderum verò in DE e$t |
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688 punctum H: dum igitur pondera AB mouentur in DE, centrum |
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689 grauitatis C ver$us D mouebitur, & ad D propius accedet; quod |
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690 e$t impo$sibile: cum pondera eandem inter $e $e $eruent di$tantiam. |
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691 Vniu$cuiu$q; enim corporis centrum grauitatis in eodem $emper <marg>2. <I>Sup. huius.</I></marg> |
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692 e$t $itu re$pectu $ui corporis. & quamquam punctum C $it duo- |
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693 rum corporum AB centrum grauitatis, quia tamen inter $e $e ita à |
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694 libra connexa $unt, vt $emper eodem modo $e $e habeant; Ideo |
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695 punctum C ita eorum erit centrum grauitatis, ac $i vna tantum |
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696 <foot>B 2</foot> |
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697 <pb> |
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698 <marg><I>Ex</I> 4. <I>primi Archim de Aequep.</I></marg> e$$et magnitudo. libra |
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699 enim vna cum ponderi- |
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700 bus vnum tantum conti |
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701 nuum efficit, cuius cen- |
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702 trum grauitatis erit $em- |
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703 per in medio. non igitur |
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704 pondus in D pondere in |
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705 E e$t grauius. Si autem |
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706 dicerent centrum graui- |
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707 tatis non in linea CD, |
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708 $ed in CE e$$e debere; |
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709 idem eueniet ab$urdum. |
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710 <fig> |
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711 <p>Amplius $i pondus D |
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712 deor$um mouebitur, pondus E $ur$um mouebit. pondus igitur gra- |
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713 uius, quàm $it E, in eodemmet $itu ponderi D æqueponderabit, & |
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714 grauia inæqualia æquali di$tantia po$ita æqueponderabunt. Adii- |
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715 ciatur ergo ponderi E aliquod graue, ita vt ip$i D contraponde- |
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716 ret, $i ex C $u$pendantur. $ed cum $upra o$ten$um $it punctum C |
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717 centrum e$$e grauitatis æqualium ponderum in DE; $i igitur pon- |
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718 <marg><I>Ex</I> 3. <I>primi Archim de Aequep.</I></marg> dus E grauius fuerit pondere D, erit centrum grauitatis in linea |
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719 CE. $itq; hoc centrum K. at per definitionem centri grauitatis, $i |
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720 pondera $u$pendantur ex K, manebunt. ergo $i $u$pendantur ex |
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721 C, non manebunt, quod e$t contra hypote$im: $ed pondus E deor |
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722 $um mouebitur. quòd $i ex C quoque $u$pen$a æqueponderarent; |
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723 <marg>1. <I>Suppo$. huius.</I></marg> vnius magnitudinis duo e$$ent centra grauitatis; quod e$t impo$si |
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724 bile. Non igitur pondus in E grauius eo, quod e$t in D, ip$i D æque- |
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725 ponderabit, cum ex puncto C fiat $u$pen$io. Pondera ergo in DE |
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726 æqualia ex eorum grauitatis centro C $u$pen$a, æqueponderabunt, |
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727 manebuntquè. quod demon$trare fuerat propo$itum. |
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728 <p><marg><I>Tartalea $exta propo $itione octa uilibri.</I></marg> Huic autem po$tremo inconuenienti ocurrunt dicentes, im- |
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729 po$sibile e$$e addere ip$i E pondus adeo minimum, quin adhuc $i |
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730 ex C $u$pendantur, pondus E $emper deor$um ver$us G moueatur. |
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731 quod nos fieri po$$e $uppo$uimus, at que fieri po$$e credebamus. ex- |
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732 ce$$um enim ponderis D $upra pondus E, cum quantitatis ratio- |
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733 nem habeat, non $olum minimum e$$e, verum in infinitum diuidi |
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734 po$$e immaginabamur, quod quidem ip$i, non $olum minimum, |
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735 <pb n=7> |
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736 $ed ne minimum quidem e$$e, cum reperiri non po$sit, hoc mo- |
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737 do demon$trare nituntur. |
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738 <fig> |
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739 <p>Exponantur eadem. |
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740 à puncti$què DE hori- |
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741 zonti perp&etilde;diculares du |
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742 cãtur DHEK, atq; alius |
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743 $it circulus LDM, cu- |
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744 ius centrũ N, qui FDG |
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745 in puncto D contingat, |
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746 ip$iq; FDG $it æqualis: |
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747 erit NC recta linea. & <marg><I>Ex</I> 12. <I>tertii.</I></marg> |
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748 quoniam angulus KEC |
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749 angulo HDN e$t æqua <marg>29. <I>Primi.</I></marg> |
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750 lis, angulusq; CEG an- |
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751 gulo NDM e$t etiam |
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752 æqualis; cum à $emidiametris, æqualibusq; circumferentiis conti- |
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753 neatur; erit reliquus mixtu$què angulus KEG reliquo mixtoquè |
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754 HDM æqualis. & quia $upponunt, quò minor e$t angulus linea |
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755 horizonti perpendiculari, & circumferentia contentus, eò pondus |
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756 in eo $itu grauius e$$e. vt quò minor e$t angulus HD, & circumfe |
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757 rentia DG contentus angulo KEG, hoc e$t angulo HDM; ita $e |
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758 cundum hanc proportionem pondus in D grauius e$$e pondere in |
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759 E. Proportio autem anguli MDH ad angulum HDG minor e$t |
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760 qualibet proportione, quæ $it inter maiorem, & minorem quanti |
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761 tatem: ergo proportio ponderum DE omnium proportionum mi |
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762 nima erit. immo neq; erit ferè proportio, cum $it omnium pro |
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763 portionum minima. quòd autem proportio MDH ad HDG $it |
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764 omnium minima, ex hac nece$sitate o$tendunt; quia MDH exce |
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765 dit HDG angulo curuilineo MDG, qui quidem angulus omnium |
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766 angulorum rectilineorum minimus exi$tit: ergo cum non po$sit da |
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767 ri angulus minor MDG, erit proportio MDH ad HDG omniũ |
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768 proportionum minima. quæ ratio inutilis valde videtur e$$e; quia |
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769 quamquam angulus MDG $it omnibus rectilineis angulis minor, |
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770 non idcirco $equitur, ab$olutè, $impliciterq; omnium e$$e angulorũ |
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771 minimum: nam ducatur à puncto D linea DO ip$i NC perpendicu |
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772 laris, hæc vtra$q; tanget circumferentias LDM FDG in puncto <marg><I>Ex</I> 18. <I>Ter tii.</I></marg> |
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773 <pb> |
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774 D. quia verò circumfe |
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775 rentiæ $unt æquales, erit |
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776 angulus MDO mixtus |
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777 angulo ODG mixto |
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778 æqualis; alter ergo an |
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779 gulus, vt ODG minor |
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780 erit MDG, hoc e$t mi |
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781 nor minimo. angulus |
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782 deinde OGH minor |
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783 erit angulo MDH; qua |
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784 re ODH ad angulum |
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785 <marg>8. <I>Quinti.</I></marg> HDG minorem habe |
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786 bit proportion&etilde;, quàm |
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787 <fig> |
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788 MDH ad eundem HDG. dabitur ergo quoquè proportio mi- |
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789 nor minima, quam in infinitum adhuc minorem ita o$tende- |
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790 mus. De$cribatur circulus DR, cuius centrum E, & $emidiame- |
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791 <marg><I>Ex</I> 11. <I>ter tit.</I></marg> ter ED. continget circumferentia DR circumferentiam DG in |
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792 <marg><I>Ex</I> 18. <I>ter tii.</I></marg> puncto D, lineamquè DO in puncto D; quare minor erit angu- |
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793 lus RDG angulo ODG. $imiliter & angulus RDH angulo |
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794 ODH. minorem igitur proportionem habebit RDH ad HDG, |
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795 quàm ODH ad HDG. Accipiatur deinde inter EC vtcun- |
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796 que punctum P, ex quo in di$tantia PD alia de$cribatur circum- |
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797 ferentia DQ, quæ circumferentiam DR, circumferentiamquè |
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798 DG in puncto D continget; & angulus QDH minor erit |
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799 angulo RDH: ergo QDH ad HDG minorem habebit propor |
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800 tionem, quàm RDH ad HDG. eodemquè pror$us modo, $i |
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801 inter PC aliud accipiatur punctum, & inter hoc &C aliud, & $ic |
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802 deinceps, infinitæ de$cribentur circum$erentiæ inter DO, & cir |
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803 cumferentiam DG; ex quibus proportionem in infinitum $emper |
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804 minorem inueniemus. atque ideo proportionem ponderis in D |
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805 ad pondus in E non adeo minorem e$$e $equitur, quin ad infini |
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806 tum ip$a $emper minorem reperiri po$sit. & quia angulus MDG |
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807 in infinitum diuidi pote$t; exce$$us quoque grauitatis D $upra E |
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808 diuidi ad infinitum poterit. |
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809 <pb n=8> |
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810 <p>Sed neque prætereundum |
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811 e$t, ip$os in demon$tratio- |
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812 ne angulum KEG maiorem |
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813 e$$e angulo HDG, tanquam |
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814 notum accepi$$e. quod e$t |
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815 quidem verum, $i DHEK |
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816 inter $e $e $int æquidi$tan- |
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817 tes. Quoniam autem (vt |
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818 ip$i quoque $upponunt) li- |
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819 neæ DHEK in centrum |
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820 mundi conueniunt; lineæ |
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821 DHEK æquidi$tantes nun |
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822 quam erunt, & angulus KEG |
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823 angirlo HDG non $olum |
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824 maior erit, $ed minor. vt |
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825 exempli gratia, producatur |
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826 FG v$que ad centrum mun |
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827 di, quod $it S; connectan- |
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828 turqué DSES. o$tenden- |
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829 dum e$t angulum SEG mi |
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830 norem e$$e angulo SDG. du |
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831 <fig> |
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832 catur à puncto E linea ET circulum DGEF contingens, ab eo |
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833 demqué puncto ip$i DS æquidi$tans ducatur EV. Quoniam igi |
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834 tur EVDS inter $e $e $unt æquidi$tantes: $imiliter ETDO æqui |
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835 di$tantes: erit angulus VET angulo SDO æqualis. & angulus |
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836 TEG angulo ODM e$t æqualis; cum à lineis contingentibus, |
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837 circumferentii$qué æqualibus contineatur: totus ergo angulus |
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838 VEG angulo SDM æqualis erit. Auferatur ab angulo SDM |
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839 angulus curuilineus MDG; ab angulo autem VEG angulus au- |
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840 feratur VES; & angulus VES rectilineus maior e$t curuilineo |
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841 MDG; erit reliquus angulus SE Gminor angulo SDG. |
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842 Quare ex ip$orum $uppo$itionibus non $olum pondus in D gra- |
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843 uius erit pondere in E; verùm è conuer$o, pondus in E ip$o D |
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844 grauius exi$tet. |
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845 <pb> |
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846 <p>Rationes tamen af |
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847 ferunt, quibus demon |
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848 $trare nituntur, libram |
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849 DE in AB horizon- |
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850 ti æquidi$tantem ex |
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851 nece$sitate redire. Pri- |
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852 mùm quidem o$ten- |
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853 dunt, idem pondus |
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854 grauius e$$e in A, |
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855 quàmin alio $itu, quem |
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856 æqualitatis $itum no- |
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857 minant, cum linea |
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858 AB $it horizonti æ- |
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859 <fig> |
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860 quidi$tans. deinde quò propius e$t ip$i A, quouis alio remotiori |
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861 grauius e$$e. Vt pondus in A grauius e$$e, quàm in D; & in D, |
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862 quàm in L. $imiliter in A grauius, quam in N; & in N grauius, |
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863 quàm in M. Vnum tantùm con$iderando pondus in altero libræ |
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864 <marg><I>Cardanus primo de $ubtilitate.</I></marg> brachio $ur$um deor$umq; moto. Quia (inquiunt) po$ita trutina |
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865 in CF, pondus in A longius e$t à trutina, quàm in D: & in D |
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866 longius, quàm in L. ductis enim DO LP ip$i CF perpendicula- |
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867 <marg><I>Ex</I> 15. <I>ter tii.</I></marg> ribus, li<*>ea AC maior e$t, quàm DO, & DO ip$a LP. quod |
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868 <marg><I>Cardanus.</I></marg> idem euenit in punctis NM. deinde ex quo loco (aiunt) pon |
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869 dus velocius mouetur, ibi grauius e$t; velocius autem ex A, quàm |
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870 ab alio $itu mouetur; ergo in A grauius e$t. $imili modo, quò |
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871 propius e$t ip$i A, velocius quoque mouetur; ergo in D gra- |
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872 <marg><I>Cardanus.</I></marg> uius erit, quàm in L. Altera deinde cau$a, quam ex rectiori, & obli |
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873 <marg><I>Iordanus propo$itio ne</I> 4.</marg> quiori motu deducunt, e$t; quò pondus in arcubus æqualibus re- |
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874 ctius de$cendit, grauius e$$e videtur; cum pondus liberum, atq; |
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875 <marg><I>Tartalea propo$itione</I> 5.</marg> $olutum $uaptè natura rectè moueatur; $ed in A rectius de$cen |
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876 dit; ergo in A grauius erit. hocq; o$tendunt accipiendo arcum |
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877 AN arcui LD æqualem; à puncti$q; NL lineæ FG (quam |
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878 etiam directionis vocant) æquidi$tantes ducantur NRLQ, quæ |
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879 lineas AB DO $ecent in QR; & à puncto N ip$i FG perpen |
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880 dicularis ducatur NT. rectèq; demon$trant LQ ip$i PO æqua |
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881 lem e$$e, & NR ip$i CT; lineamq; NR ip$a LQ maiorem e$$e. |
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882 Quoniam autem de$cen$u; ponderis ex A v$q; ad N per circum- |
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883 <pb n=9> |
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884 ferentiam AN maiorem portionem lineæ FG pertran$it (quod |
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885 ip$i vocant capere de directo) quàm de$cen$us ex L in D per cir |
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886 cumferentiam LD; cùm de$cen$us AN lineam CT pertran$eat, |
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887 de$cen$us verò LD lineam PO; & CT maior e$t PO; rectior erit |
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888 de$cen$us AN, quám de$cen$us LD. grauius ergo erit pondus |
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889 in A, quàm in L, & in quouis alio $itu. eodemq; pror$us |
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890 modo o$tendunt, quò propius e$t ip$i A, grauius e$$e. |
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891 Vt $int circumferentiæ LD DA inter $e $e æquales, & à puncto |
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892 Dip$i AB perpendicularis ducatur DR; erit DR ip$i CO æqua <marg>34 <I>Primi.</I></marg> |
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893 lis. lineam deinde DR ip$a LQ maiorem e$$e demon$trant. di- |
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894 cuntq; de$cen$um DA magis capere de directo de$cen$u LD, ma |
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895 ior enim e$t linea CO, quàm OP; quare pondus grauius erit |
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896 in D, quàm in L. quod ip$um euenit in punctis NM. Suppo- |
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897 $itionem itaq;, qua libram DE in AB redire demon$trant, vt <marg><I>Iordanus $uppo$itione</I> 4.</marg> |
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898 notam, manife$tamq; proferunt. Nempè Secundùm $itum pon |
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899 dus grauius e$$e, quanto in eodem $itu minus obliquus e$t de$cen |
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900 $us. huiu$q; reditus cau$am eam e$$e dicunt; Quoniam $cilicet <marg><I>Iordanus propo$itio ne</I> 3.</marg> |
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901 de$cen$us ponderis in D rectior e$t de$cen$u ponderis in E, cùm |
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902 minus capiat de directo pondus in E de$cendendo, quàm pon <marg><I>Tartalea propo$itio ne</I> 5.</marg> |
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903 dus in D $im liter de$cendendo. Vt $i arcus EV $it ip$i DA |
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904 æqualis, ducanturq; VH ET ip$i FG perpendiculares; maior |
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905 erit DR, quàm TH. quare per $uppo$itionem pondus in D ra |
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906 tione $itus grauius erit pondere in E. pondus ergo in D, cùm $it |
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907 grauius, deor$um mouebitur; pondus verò in E $ur$um, donec li |
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908 bra DE in AB redeat. |
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909 <p>Altera huius quoq; reditus ratio e$t, cùm trutina $upra libram <marg><I>Cardanus.</I></marg> |
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910 e$t in CF; linea CG e$t meta. & quoniam angulus GCD ma |
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911 ior e$t angulo GCE, & maior à meta angulus grauius reddit |
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912 pondus; trutina igitur $uperius exi$tente, grauius erit pondus in |
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913 D, quàm in E. idcirco D in A, & E in B redibit. |
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914 <p>His itaq; rationibus conantur o$tendere libram DE in AB re |
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915 dire; quæ meo quidem iuditio facile $olui po$$unt. |
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916 <foot>C</foot> |
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917 <pb> |
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918 <p>Primùm itaq; quan |
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919 tum attinet ad ratio- |
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920 nes pondus in A gra |
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921 uius e$$e, quàm in a- |
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922 lio $itu o$tendentes, |
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923 quas ex longiori, & |
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924 propinquiori di$tãtia à |
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925 linea FG, & ex velo- |
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926 ciori, & rectiori mo |
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927 tu à puncto A dedu- |
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928 cunt; primùm quidem |
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929 non demon$trant, cur |
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930 pondus ex A velocius |
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931 <fig> |
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932 moueatur, quàm ex alio $itu. nec quia CA e$t DO maior, |
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933 & DO ip$a LP, propterea $equitur tanquam ex vera cau$a, pon |
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934 dus in A grauius e$$e, quàm in D; & in D, quàm in L. neq; |
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935 enim intellectus quie$cit, ni$i alia huius o$tendatur cau$a; cùm po |
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936 tius $ignum, quàm vera cau$a e$$e videatur. id ip$um quoq; al- |
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937 teri rationi contintingit, quam ex rectiori & obliquiori motu de- |
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938 ducunt. Præterea quæcunq; ex velociori, & rectiori motu per- |
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939 $uadent pondus in A grauius e$$e, quàm in D; non ideo de- |
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940 mon$trant pondus in A, quatenus e$t in A, grauius e$$e pon |
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941 dere in D, quatenus e$t in D; $ed quatenus à punctis DA rece |
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942 dit. Idcirco antequàm vlterius progrediar, o$tendam primùm |
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943 pondus, quò propius e$t ip$is FG, minus grauitare; tum qua- |
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944 tenus in eo $itu, in quo reperitur, manet: tum quatenus ab eo |
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945 recedit. $imulq; fal$um e$$e, pondus in A grauius e$$e, quàm in |
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946 alio $itu. |
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947 <pb n=10> |
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948 <p>Producatur FG v$q; ad mun di con |
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949 trum, quod $it S. & à puncto S circu |
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950 lum AFBG contingens ducatur. neq; |
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951 enim linea à puncto S circulum con- |
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952 tingere pote$t in A; nam ducta AS |
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953 triangulum ACS duos haberet angu |
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954 los rectos, nempè SAC ACS, quod <marg>18 <I>Tertii.</I></marg> |
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955 e$t impo$sibile. neq; $upra punctum A |
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956 in circumferentia AF continget; cir |
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957 culum enim $ecatet. tanget igitur in- |
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958 fra, $itq; SO. connectantur deinde SD |
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959 SL, quæ circumferentiam AOG in |
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960 punctis KH $ecent. & Ck CH con |
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961 iungantur. Et quoniam pondus, quanto |
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962 propius e$t ip$i F, magis quoque inni- |
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963 titur centro; vt pondus in D magis ver- |
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964 $ionis puncto C innititur tanquam |
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965 centro; hoc e$t in D magis $upra li- |
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966 neam CD grauitat, quàm $i e$$et in A |
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967 $upralineam CA; & adhuc magis in |
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968 L $upra lineam CL; Nam cùm tres |
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969 anguli cuiu$cunq; trianguli duobus re- |
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970 <fig> |
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971 ctis $int æquales, & trianguli DCk æquicruris angulus DCk |
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972 minor $it angulo LCH æquicruris trianguli LCH: erunt reli- |
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973 qui ad ba$im $cilicet CDk CkD $imul $umpti reliquis CLH |
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974 CHL maiores. & horum dimidii; hoc e$t angulus CDS angu |
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975 lo CLS maior erit. cùm itaq; CLS $it minor, linea CL ma |
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976 gis adhærcbit motui naturali ponderis in L pror$us $oluti. hoc |
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977 e$t lineæ LS, quàm CD motui DS. pondus enim in L libe- |
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978 berum, atq; $olutum in centrum mundi per LS moueretur, pon- |
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979 dusq; in D per DS. quoniam verò pondus in L totum $uper LS |
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980 grauitat, in D verò $uper DS: pondus in L magis $upra lineam |
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981 CL grauitabit, quàm exi$tens in D $upra lineam DC. ergo |
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982 linea CL pondus magis $u$tentabit, quàm linea CD. Eodem- |
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983 qué modo, quò pondus propiu fuerit ip$i F, magis ob hanc cau- |
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984 $am à linea CL $u$tineri o$tendetur-$emper enim angulus CLS |
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985 <foot>C 2</foot> |
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986 <pb> |
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987 minor e$$et. quod etiam patet; quia $i |
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988 lineæ CL, & LS in vnam coinciderent |
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989 lineam, quod euenit in FCS; tunc linea |
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990 CF totum $u$tineret pondus in F, im- |
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991 mobilemq; redderet: neq; vllam pror- |
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992 $us grauitatem in circumferentia circu- |
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993 li haberet. Idem ergo pondus propter |
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994 $ituum diuer$itatem grauius, leuiu$q; erit. |
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995 non autem quia ratione $itus interdum |
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996 maiorem re vera acquirat grauitatem, |
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997 interdum verò amittat, cùm eiu$dem $it |
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998 $emper grauitatis, vbicunque reperiatur; |
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999 $ed quia magis, minu$uè in circumferen- |
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1000 tia grauitat, vt in D magis $upra circum |
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1001 ferentiam DA grauitat, quàm in L $upra |
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1002 circumferentiam LD. hoc e$t, $i pon |
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1003 dus à circumferentiis, recti$q; lineis $u |
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1004 $tineatur; circumferentia AD magis $u |
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1005 $tinebit pondus in D, quàm circumfe |
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1006 rentia DL pondere exi$tente in <I>L.</I> mi |
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1007 nus enim coadiuuat CD, quàm CL. |
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1008 Præterea quando pondus e$t in L, $i e$- |
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1009 <fig> |
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1010 $et omnino liberum, penitu$q; $olutum, deor$um per LS moueretur; |
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1011 ni$i à linea CL prohiberetur, quæ pondus in L vltra lineam LS per |
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1012 circumferentiã LD moueri cogit; ip$umq; quodammodo impellit, |
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1013 impellendoq; pondus partim $u$tentabit. ni$i enim $u$tineret, ip$iq; |
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1014 reniteretur, deor$um per lineam LS moueretur, non autem per |
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1015 circumferentiam LD. $imiliter CD ponderi in D renititur, cùm |
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1016 illud per circumferentiam DA moucri cogat. eodemq; modo |
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1017 exi$terte pondere in A, linea CA pondus vltra lineam AS per |
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1018 circumferentiam AO moueri compellet. e$t enim angulus CAS |
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1019 acutus; cùm angulus ACS $it rectus. lineæ igitur CA CD ali |
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1020 qua ex parte, non tamen ex æquo ponderi renituntur. & quotie$ |
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1021 cunque angulus in circumferentia circuli à lineis à centro |
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1022 mundi S, & centro C prodeuntibus, fuerit acutus; idem eue- |
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1023 nire $imiliter o$tendemus. Quoniam autem mixtus angulus CLD |
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1024 <pb n=11> |
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1025 æqualis e$t angulo CDA, cùm à femidiametris, eademq; circumfe |
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1026 rentia contineantur; & angulus C<I>L</I>S angulo CDS e$t minor; |
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1027 erit reliquus <I>s</I>LD reliquo SDA maior. quare circumferentia |
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1028 DA, hoc e$t de$cen$us ponderis in D propior erit motui natu- |
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1029 rali ponderis in D $oluti, lineæ $cilicet DS, quàm circumferen |
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1030 tia LD lineæ LS. minus igitur linea CD ponderi in D reniti- |
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1031 tur, quàm linea CL ponderi in L. linea ideo CD minus $u$tinet, |
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1032 quàm CL; pondu$q; magis liberum erit in D, quàm in L: |
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1033 cùm pondus naturaliter magis per DA moueatur, quàm per LD. |
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1034 quare grauius erit in D, quàm in L. $imiliter o$tendemus CA |
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1035 minus $u$tinere, quàm CD: pondu$q; magis in A, quàm in Dli |
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1036 berum, grauiu$q, e$$e. Ex parte deinde inferiori ob ea$dem cau$as, |
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1037 quò pondus propius fuerit ip$i G, magis detinebitur, vt in Hma |
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1038 gis à linea CH, quàm in K à linea CK. nam cùm angulus CHS |
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1039 maior $it angulo CkS, ad rectitudinem magis appropinquabunt <marg>21 <I>primi.</I></marg> |
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1040 $e $e lineæ CHHS, quàm Ck kS; atq; obid pondus magis deti- |
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1041 nebitur à CH, quàm à Ck $i enim CH HS in vnam conuenirent |
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1042 lineam vt euenit pondere exi$tente in G; tunc linea CG to tum $u |
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1043 $tineret' pondus in G, ita vt immobilis per$i$teret. quò igitur |
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1044 minor erit angulus linea CH, & de$cen$u ponderis $oluti, $cilicet |
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1045 HS contentus, eò minus quoq; eiu$modi linea pondus detinebit. |
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1046 & vbiminus detinebitur, ibi magis liberum, grauiu$q; exi$tet. |
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1047 Præterea $i pondus in k liberum e$$et, atq; $olutum, per lineam |
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1048 k S moueretur; à linea verò Ck prohibetur, quæ cogit pondus |
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1049 citrà lineam k S per circumferentiam k H moueri. ip$um enim |
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1050 quodammodo retrahit, retrahendoq; $u$tinet. ni$i enim $u$tineret. |
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1051 pondus deor$um per rectam k S moueretur, non autem per cir |
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1052 cumferentiam k H. $imiliter CH pondus retinet, cùm per circum |
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1053 ferentiã HG moueri compellat. Quoniã autem angulus CHS ma- |
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1054 ior e$t angulo CKS, d&etilde;ptis æqualibus angulis CHG CkH; erit |
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1055 reliquus SHG reliquo SKH maior. circumferentia igitur k H, hoc |
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1056 e$t de$cen$us ponderis in k, propior erit motui naturali ponderis in |
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1057 k $oluti, hoc e$t lineæ k S, quàm circumferentia HG lineæ HS. mi |
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1058 nus idcirco detinet linea Ck, quàm CH: cùm pondus naturali- |
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1059 ter magis moueatur per k H, quàm per HG. $imiliratione o$ten- |
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1060 detur, quò minor erit angulus SkH, lineam Ck minus $u$tinere. |
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1061 <pb> |
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1062 exi$tente igitur pondere in O, quia angu |
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1063 lus SOC non $olum minor e$t angulo |
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1064 CKS, verùm etiam omnium angulorum |
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1065 à punctis CS prodeuntium, verticemq; |
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1066 in circumferuntia OkG habentium mi- |
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1067 nimus; erit anglus SOK, & angulo SkH, |
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1068 & eiu$modi omnium minimus. ergo de- |
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1069 $cen$us ponderis in O propior erit motui |
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1070 naturali ip$ius in O $oluti, quàm in alio |
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1071 $itu circumferentiæ OkG. lineaq; CO |
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1072 minus pondus $u$tinebit, quàm $i pon- |
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1073 dusin quouis alio fuerit $itu eiu$dem cir |
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1074 cumferentiæ OG. $imiliter quoniam con |
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1075 tingentiæ angulus SOk, & angulo SDA, |
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1076 & SAO, ac quibu$cunq; $imilibus e$t mi |
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1077 nor; erit de$cen$us ponderis in O motui |
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1078 naturali ip$ius ponderis in O $oluti pro- |
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1079 pior, quàm in alio $itu circumferentiæ |
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1080 ODF. Præte reaquoniam linea GO pon |
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1081 dus in O dum deor$um mouetur, impelle- |
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1082 re nonpote$t, ita vt vltra lineam OS mo |
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1083 ueatur; cùm linea OS circulum non $ecet, |
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1084 <fig> |
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1085 $ed contingat; angulu$q; SOC $it rectus, & non acutus; pondus |
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1086 in O nihil $upra lineam CO grauitabit. neq; centro innitetur. quem |
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1087 admodum in quouis alio puncto $upra O accideret. erit igitur pon |
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1088 dus in O magis ob has cau$as liberum, atq; $olutum in hoc $itu, |
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1089 quàm in quouis alio circumferentiæ FOG. acidcirco in hoc |
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1090 grauius erit, hoc e$t magis grauitabit, quàm in alio $itu. & quò |
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1091 propius fuerit ip$i O remotiori grauius erit. lineaq; CO horizonti |
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1092 æquidi$tans erit. non tamen puncti C horizonti (vt ip$i exi$ti- |
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1093 mant) $ed ponderis in O con$tituti, cùm ex centro grauitatis |
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1094 ponderis $ummendus $it horizon. quæ omnia demon$trare opor- |
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1095 tebat. |
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1096 <pb n=12> |
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1097 <p>Si autem libræ brachium ip$o CO |
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1098 fuerit maius, putá quantitate CD; erit |
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1099 quoq; pondus in O grauius. circulus de- |
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1100 $cribatur OH, cuius centrum $it D, $e <marg><I>Ex</I> 11 <I>Ter tit.</I></marg> |
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1101 midiameterq; DO. tanget circulus OH |
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1102 circulum FOG in puncto O, lineamq; <marg><I>Ex</I> 18 <I>Ter tii.</I></marg> |
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1103 OS, quæ ponderis in O rectus, natura- |
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1104 li$q; e$t de$cen$us, in eodem puncto con |
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1105 tinget. & quoniam angulus SOH mi- |
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1106 nor e$t angulo SOG, erit de$cen$us |
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1107 ponderis in O per circumferentiam OH |
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1108 motui naturali OS propior, quàm per |
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1109 circumferentiam OG. magis ergo li- |
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1110 berum, atq; $olutum, ac per con$equens |
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1111 grauius erit in O, centro libræ exi$ten |
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1112 te in D, quàm in C. $imiliter o$ten- |
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1113 detur, quò maius fuerit brachium DO, |
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1114 pondus in O adhuc grauius e$$e. |
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1115 <fig> |
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1116 <pb> |
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1117 <p>Siverò idem circulus AFBG, |
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1118 cuius centrum $it R, propius fuerit |
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1119 mundi centro S; circulumqué à pun- |
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1120 cto S ducatur contingens ST; punctum |
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1121 T (vbi grauius e$t pondus) magis |
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1122 à puncto A di$tabit, quàm punctum |
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1123 O. ducantur enim à punctis OT ip$i |
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1124 CS perpendiculares OMTN; conne |
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1125 ctanturq; RT; $itq; centrum R in li- |
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1126 nea CS; lineaq; ARB ip$i ACB æqui |
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1127 <marg><I>Cor.</I> 8 <I>$exti</I></marg> di$tans. Quoniam igitur triangula COS |
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1128 RTS $unt rectangula; erit SC ad CO, |
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1129 vt CO ad CM. $imiliter SR ad RT, |
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1130 vt RT ad RN. cùm itaq; $it RT ip- |
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1131 <marg><I>Ex</I> 8 <I>quinti</I></marg> $i CO æqualis, & SC ip$a SR maior: |
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1132 maiorem habebit proportionem SC |
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1133 ad CO, quàm SR ad RT. quare ma |
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1134 iorem quoq; proportionem habebit |
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1135 CO ad CM, quàm RT ad RN. mi |
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1136 <marg><I>Ex</I> 10 <I>quin ti.</I></marg> nor ergo erit CM, quàm RN. $ecetur |
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1137 igitur RN in P, ita vt RP $it ip$i |
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1138 <fig> |
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1139 CM æqualis; & à puncto P ip$is MONT æquidi$tans ducatur |
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1140 PQ, quæ circumferentiam AT $ecet in Q: deniq; connectatur |
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1141 RQ. quoniam enim duæ CO CM duabus RQRP $unt æqua |
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1142 <marg>7 <I>Sexti.</I></marg> les, & angulus CMO angulo RPQ e$t æqualis; erit & angu- |
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1143 lus MCO angulo PRQ æqualis. angulus autem MCA rectus |
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1144 <marg>26 <I>Tertii.</I></marg> recto PRA e$t æqualis; ergo reliquus OCA reliquo QRA |
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1145 æqualis, & circumferentia OA circumferentiæ QA æqualis quo- |
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1146 que erit. punctum idcirco T, quia magis à puncto A di$tat, |
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1147 quàm Q; magis quoq; à puncto A di$tabit, quàm punctum O. |
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1148 $imiliter o$tendetur, quò propius fuerit circulus mundi centro, eun- |
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1149 dem magis di$tare. atq; ita vt prius demon$trabitur pondus in cir |
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1150 cumferentia TAF centro R inniti, in circumferentia verò TG |
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1151 à linea detineri; atq; in puncto T grauius e$$e. |
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1152 <pb n=13> |
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1153 <p>Si autem punctum G e$$et |
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1154 in centro mundi; tunc quò |
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1155 pondus propius fuerit ip$i G, |
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1156 grauius erit: & vbicunq; po |
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1157 natur pondus præterquàm in |
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1158 ip$o G, $emper centro C inni |
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1159 tetur, vt in K. nam ducta |
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1160 G k, efficiet hæc ($ecun- |
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1161 dùm quam fit ponderis natu |
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1162 ralis motus) vná cum libræ |
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1163 brachio k C angulum acu- |
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1164 tum. æquicruris enim trian- |
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1165 guli CkG ad ba$im anguli |
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1166 ad k, & G $unt $emper acuti. |
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1167 <fig> |
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1168 Conferantur autem inuicem hæc duo, pondus videlicet in k, & |
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1169 pondus in D: erit pondus in k grauius, quàm in D. nam iuncta |
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1170 DG, cùm tres anguli cuiu$cunque trianguli duobus $int rectis |
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1171 æquales, & trianguli CDG æquicruris angulus DCG maior $it |
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1172 angulo kCG æquicruris trianguli CkG: erunt reliqui ad ba$im an |
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1173 guli DGC GDC $imul $umpti reliquis KGCGkC $imul $umptis |
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1174 minores. horumq; dimidii; angulus $cilicet CDG angulo CKG |
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1175 minor erit. quare cùm pondus in k $olutum naturaliter per |
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1176 KG moueatur, pondusq; in D per DG, tanquam per $patia, |
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1177 quibus in centrum mundi feruntur; linea CD, hoc e$t libræ |
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1178 brachium magis adhærebit motui naturali ponderis in D pror- |
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1179 $us $oluti, lineæ $cilicet DG; quàm Ck motui $ecundùm kG |
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1180 effecto. magis igitur $u$tinebit linea CD, quàm Ck. ac pro- |
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1181 pterea pondus in k ex $uperius dictis grauius erit, quàm in D. |
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1182 Præterea quoniam pondus in K $i e$$et omnino liberum, pror$u$q; |
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1183 $olutum, deor$um per k G moueretur; ni$i à linea C k prohibere |
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1184 tur, quæ pondus vltra lineam KG per circumferentiam KH mo- |
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1185 ueri cogit; linea C k pondus partim $u$tinebit, ip$iq; renitetur; |
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1186 cùm illud per circumferentiam k H moueri compellat. & |
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1187 quoniam angulus CDG minor e$t angulo CkG, & angulus CDk |
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1188 angulo CkH e$t æqualis; crit reliquus GDk reliquo G k H maior. |
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1189 circumferentia igitur k H motui naturali ponderis in k $oluti, li- |
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1190 <foot>D</foot> |
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1191 <pb> |
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1192 neæ $cilicet KG propior erit, |
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1193 quàm circumferentia Dk li- |
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1194 neæ DG. quare linea CD |
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1195 ponderi in D magis renititur, |
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1196 quàm linea C k ip$i ponde- |
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1197 ri in K. ergo pondus in k |
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1198 grauius erit, quàm in D. |
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1199 Similiter o$tendetur pondus, |
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1200 quò fuerit ip$i F propius, vt |
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1201 in L, minus grauitare: pro- |
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1202 pius verò ip$i G, vt in H, |
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1203 grauius e$$e. |
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1204 <fig> |
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1205 <p>Si verò centrum mundi |
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1206 S e$$et inter puncta CG; |
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1207 primùm quidem $imili- |
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1208 ter o$tendetur pondus vbi |
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1209 cunq; po$itum centro C |
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1210 initi, vt in H. ductis enim |
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1211 HG HS, angulus ad |
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1212 ba$im GHC æquicruris tri |
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1213 anguli CHG e$t $emper |
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1214 acutus: quare & SHC ip |
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1215 $o minor erit quoq; $em |
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1216 per acutus. ducatur au- |
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1217 tem à puncto S ip$i CS |
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1218 perpendicularis Sk. di- |
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1219 <fig> |
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1220 co pondus grauius e$$e in k, quàm in alio $itu circumferentiæ FKG. |
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1221 & quò propius fuerit ip$i F, vel G, minus grauitare. Accipiantur |
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1222 ver$us F puncta DL, connectanturq; LC LS DC DS, produ- |
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1223 canturq; LS DS k SHS v$q; ad circuli circumferentiam in EM |
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1224 NO; connectanturq; CE, CM, CN, CO. Quoniam enim |
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1225 <marg>35 <I>Tertii.</I></marg> LE DM $e inuicem $ecant in S; erit rectangulum LSE rectan- |
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1226 <marg>16 <I>Sexti.</I></marg> gulo DSM æquale. quare vt LS ad DS ita erit SM |
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1227 <marg>7 <I>Tertii.</I></marg> ad SE. maior autem e$t LS, quàm DS; & SM ip$a SE. |
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1228 <pb n=14> |
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1229 ergo LS SE $imul $umptæ ip$is DS SM maiores erunt. eademq; <marg>25 <I>Quinti.</I></marg> |
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1230 ratione kN minorem e$$e DM o$tendetur. rur$us quoniamre |
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1231 ctangulum OSH æquale e$t rectangulo kSN; ob eandem cau$am |
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1232 HO maior erit kN. eodemq; pror$us modo kN omnibus a- |
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1233 liis per punctum S tran$euntibus minorem e$$e demon$trabitur. |
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1234 & quoniam æquicrurium triangulorum CLE DCM latera LC |
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1235 CE lateribus DC CM $unt æqualia; ba$is verò LE maior e$t |
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1236 DM: erit angulus LCE angulo DCM maior. quare ad ba$im <marg>25 <I>Primi.</I></marg> |
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1237 anguli C<I>L</I>E CEL $imul $umpti angulis CDM CMD mi- |
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1238 nores erunt. & horum dimidii, angulus $cilicet CLS angulo CDS |
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1239 minor erit. ergo pondus in <I>L</I> magis $upra lineam LC, quàm |
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1240 in D $upra DC grauitabit, magisqué centro innitetur in L, quàm |
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1241 in D. $imiliter o$tendetur in D magis c&etilde;tro Cinniti, quàm in k. ergo |
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1242 ponds in k grauius erit, quàm in D; & in D, quàm in L. eademq; pror |
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1243 $us ratione quoniam kN minor e$t HO, erit angulus CKS an- |
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1244 gulo CHS maior. quare pondus in H magis centro C innite- |
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1245 tur, quàm in k. & hoc modo o$tendetur, vbicunq; in circum- |
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1246 ferentia FDG fuerit pondus, minus in K centro C inniti, quàm |
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1247 in alio $itu: & quò propius fuerit ip$i F, vel G, magis inniti. dein- |
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1248 de quoniam angulus CkS maior e$t CDS, & CDk æqualis |
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1249 e$t CkH: erit reliquus SkH reliquo SDk minor. quare cir- |
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1250 cumferentia k H propior erit motui naturali recto ponderis in K |
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1251 $oluti, lineæ $cilicet k S, quàm circumferentia D k motui DS. & |
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1252 ideo linea CD magis ip$i ponderi in D renititur, quàm CK |
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1253 ponderi in k con$tituto. hacq; ratione o$tendetur angulum |
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1254 SHG maiorem e$$e SkH: & per con$equens lineam CH magis |
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1255 ponderi in H reniti, quàm CK ponderi in K. $imiliter demon- |
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1256 $trabitur lineam C<I>L</I> magis pondus $u$tinere, quàm CD: ob |
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1257 ea$demq; cau$as o$tendetur pondus in K minus $upra lineam Ck |
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1258 grauitare, quàm in quouis alio $itu fuerit circumferentiæ FDG. |
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1259 & quò propius fuerit ip$i F, vel G, minus grauitare. grauius ergo |
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1260 erit in k, quàm in alio $itu: minu$q; graue erit, quò propius fue- |
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1261 rit ip$i F. vel G. |
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1262 <foot>D 2</foot> |
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1263 <pb> |
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1264 <p>Si deniq; centrum C |
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1265 e$$et in centro mundi, |
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1266 pondus vbicunque con- |
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1267 $titutum manere mani- |
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1268 fe$tum e$t. vt po$ito pon |
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1269 dere in D, linea CD to- |
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1270 tum $u$tinebit pondus; |
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1271 cùm ip$ius ponderis in D |
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1272 horizonti $it perpendicu |
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1273 <marg>1 <I>Huius.</I></marg> laris. pondus ergo ma |
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1274 nebit. |
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1275 <fig> |
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1276 <p>Quoniam autem in his hactenus demon$tratis, nullam de gra |
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1277 uitate brachii libræ mentionem fecimus, idcirco $i brach$i quoq; |
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1278 grauitatem con$iderare voluerimus, centrum grauitatis magnitu |
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1279 dinis ex pondere, brachioq; compo$itæ inueniri poterit, circulo |
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1280 rumq; circumferentiæ $ecundum di$tantiam à centro libræ ad |
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1281 hoc ip$um grauitatis centrum de$cribentur, ac $i in ip$o (vt re ue |
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1282 ra e$t) pondus con$titutum fuerit; omnia, $icuti ab$q; libræ bra |
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1283 chii grauitate con$iderata inuenimus; hoc quoq; modo eius con$i |
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1284 derata grauitate reperiemus. |
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1285 <pb n=15> |
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1286 <p>Ex dictis igitur, con$iderando li- |
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1287 bram, vt longè à mundi centro a- |
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1288 be$t, quemadmodum ip$i fecere, $i- |
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1289 cuti etiam actu e$t, apparet fal$itas |
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1290 dicentium pondus in A grauius e$$e, |
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1291 quàm in alio $itu. $imulq; fal$um e$$e, |
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1292 quò pondus à linea FG magis di$tat |
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1293 grauiuis e$$e. nam punctum O pro- |
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1294 pius e$t ip$i FG, quàm punctum A. |
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1295 e$t enim linea à puncto O ip$i FG <marg><I>Ex</I> 15 <I>Tertii.</I></marg> |
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1296 perpendicularis ip$a CA minor. de- |
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1297 inde ex puncto A pondus velocius mo |
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1298 ueri, quàm ab alio $itu, e$t quoque |
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1299 fal$um. ex puncto enim O pondus ve- |
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1300 locius mouebitur, quàm ex puncto |
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1301 A; cùm in O $it magis liberum, atq; |
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1302 $olutum, quàm in alio $itu: de$cen$us |
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1303 qué ex puncto O propior $it motui na- |
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1304 turali recto, quàm quilibet alius de- |
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1305 $cen$us. |
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1306 <fig> |
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1307 <p>Præterea cùm ex re- |
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1308 ctiori, & obliquiori defc&etilde; |
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1309 $u o$tendunt, pondus in |
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1310 A grauiur e$$e, quàm in |
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1311 D; & in D, quàm in |
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1312 L; primùm quidem fal |
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1313 $um exi$timant, $i pon |
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1314 dus aliquod collocatum |
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1315 fuerit in quocunq; $itu |
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1316 circunferentiæ, vt in D, |
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1317 rectum eius de$cen$um |
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1318 per rectam lineam DR |
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1319 ip$i FG parallelam, tam |
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1320 quàm $ecundùm mo- |
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1321 <fig> |
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1322 <pb> |
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1323 tum naturalem fieri de- |
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1324 bere; $icuti prius dictum |
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1325 e$t. In quocunq; enim |
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1326 $itu pondus aliquod con |
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1327 $tituatur, $i naturalem |
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1328 eius ad propium locum |
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1329 motionem $pectemus, |
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1330 cùm rectá ad eum $ua- |
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1331 ptè natura moueatur, $up |
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1332 po$ita totius vniuer$i figu |
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1333 ra, eiu$modi erit; vt |
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1334 $emper $patiũ, per quod |
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1335 naturaliter mouetur, ra- |
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1336 tionem habere videatur |
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1337 <fig> |
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1338 lineæ à circumferentia ad centrum productæ. non igitur natura |
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1339 les de$cen$us recti cuiuslibet $oluti ponderis per lineas fieri po$ |
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1340 $unt inter $e $e parallelas; cùm omnes in centrum mundi conue- |
|
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1341 niant. $upponunt deinde ponderis ex Din A per rectam lineam |
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1342 ver$us centrum mundi motum eiu$dem e$$e quantitatis, ac $i fui$ |
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1343 $et ex O in C: ita vt punctum A æqualiter à centro mundi $it |
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1344 di$tans, vt C. quod e$t etiam fal$um; nam punctum A magis |
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1345 à centro mundi di$tat, quàm C: maior enim e$t linea à cen- |
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1346 <marg>18 <I>Primi.</I></marg> tro mundi v$q; ad A, quàm à centro mundi v$q; ad C: cùm li- |
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1347 nea à centro mundi v$q; ad A rectum $ubtendat angulum à li- |
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1348 neis AC, & à puncto C ad centrum mundi contentum. ex qui- |
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1349 bus non $olum $uppo$itio illa, qua libram DE in AB redire demon |
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1350 $trant, verùm etiam omnes ferè ip$orum demon$trationes ruunt. |
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1351 ni$i forta$$e dixerint, hæc omnia propter maximam à centro mun |
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1352 di v$q; ad nos di$tantiam adeo in$en$ibilia e$$e, vt propter in$en |
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1353 $ibilitatem tanquam vera $upponi po$sint: cùm omnes quid&etilde; alii, qui |
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1354 hæc tractauerunt, tanquam nota $uppo$uerint. præ$ertim quia |
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1355 $en$ibilitas illa non efficit, quin de$cen$us ponderis ex L in D |
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1356 (vt eorum verbis vtar) minus capiat de directo, quàm de$cen- |
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1357 $us DA. $imiliter arcus DA magis de directo capiet, quàm cir |
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1358 cumferentia EV. quocirca vera erit $uppo$itio; aliæq; demon- |
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1359 $trationes in $uo robore permanebunt. Concedamus etiam pon |
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1360 <pb n=16> |
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1361 dus in A grauius e$$e, quàm in alio $itu; rectumq; ponderis de- |
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1362 $cen$um per rectam lineam ip$i FG parallelam fieri debere; & |
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1363 quælibet puncta in lineis horizonti æquidi$tantibus accepta æ- |
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1364 qualiter à centro mundi di$tare: non tamen propterea $equetur, |
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1365 veram e$$e demon$trationem, qua inferunt pondus in A grauius |
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1366 e$$e, quàm in alio $itu, vt in L. $i enim verum e$$et, quò pon |
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1367 dus hoc modo rectius de$cendit, ibi grauius e$$e; $equeretur etiam, |
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1368 quò idem pondus in æqualibus arcubus æqualiter rectè de$cende |
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1369 ret, vt in ii$dem locis æqualem haberet grauitatem, quod fal |
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1370 $um e$$e ita demon$tratur. |
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1371 <p>Sint circumferentiæ AL AM inter $e $e æquales; & conne |
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1372 ctatur LM, quæ AB $ecet in X: erit LM ip$i FG æquidi$tans, |
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1373 ip$iq; AB perpendicularis. & XM ip$i XL æqualis erit. $i igi <marg><I>Ex</I> 3 <I>Tertii.</I></marg> |
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1374 tur pondus ex L moueatur in A per circumferentiam LA, rectus |
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1375 eius motus erit $ecundùm lineam LX. $i verò moueatur ex A |
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1376 in M per circum$erentiam AM, $ecundùm rectam eius motus |
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1377 erit XM. quare de$cen$us ex L in A æqualis erit de$cen$ui ex A |
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1378 in M; tum ob circumferentias æquales, tum propter rectas li |
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1379 neas ip$i AB perpendiculares æquales. ergo idem pondus in L |
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1380 æquè graue erit, vt in A, quod e$t fal$um. cum longé grauius $it |
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1381 in A, quàm in L. |
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1382 <p>Quamuis autem AMLA æqualiter $ecundùm ip$os de directo |
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1383 capiant; dicent forta$$e, quia tamen principium de$cen$us ex L |
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1384 $cilicet LD minus de directo capit, quàm principium de$cen$us |
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1385 ex A, $cilicet AN; pondus in A grauius erit, quàm in L. nam |
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1386 cùm circumferentia AN $it ip$i LD (vt $upra po$itum e$t) |
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1387 æqualis, quæ $ecundùm ip$os de directo capit CT; LD verò |
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1388 de directo capit PO. ideo pondus grauius erit in A, quàm in L. |
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1389 quod $i verum e$$et, $equeretur idem pondus in eodem $itu diuer |
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1390 $o duntaxat modo con$ideratum in habitudine ad eundem $itum, |
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1391 tum grauius, tum leuius e$$e. quod e$t impo$sibile. hoc e$t, $i |
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1392 de$cen$um con$ideremus ponderis in L, quatenus ex L in A de- |
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1393 $cendit, grauius erit, quàm $i eiu$dem ponderis de$cen$um con- |
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1394 $ideremus ex L in D tantùm. neq; enim negare po$$unt ex ei$- |
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1395 demmet dictis, quin de$cen$us ponderis ex L in A de directo ca |
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1396 piat LX, $iue PC. de$cen$us verò AM, quin $imiliter de directo |
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1397 <pb> |
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1398 capiat XM: cùm ip$i |
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1399 quoq; hoc modo acci- |
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1400 piant, atq; ita accipe- |
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1401 re $it nece$$e. $i enim li- |
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1402 bram DE in AB redire |
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1403 demon$trare volunt, com |
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1404 parando de$cen$us pon- |
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1405 deris in D cum de$cen- |
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1406 $u ponderis in E, nece$$e |
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1407 e$t, vt o$tendant rectum |
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1408 de$cen$um OC corre- |
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1409 $pondentem circumferen |
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1410 tiæ DA maiorem e$$e re |
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1411 cto de$cen$u TH circum |
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1412 <fig> |
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1413 ferentiæ EV corre$pondente. $i enim partem tantùm totius de- |
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1414 $cen$us ex D in A acciperent, vt D k; o$tenderentq; magis cape- |
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1415 re de directo de$cen$um Dk, quàm æqualis portio de$cen$us ex |
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1416 puncto E. $equetur pondus in D $ecundùm ip$os grauius e$$e pon |
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1417 dere in E; & v$q; ad k tantùm deor$um moueri: ita vt libra mo |
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1418 ta $it in kI. $imiliter $i libram KI in AB redire demon$trare vo |
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1419 lunt accipiendo portionem de$cen$us ex k in A; hoc e$t k S; |
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1420 o$tenderentq; k S magis de directo capere, quàm ex aduer$o æ- |
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1421 qualis de$cen$us ex puncto I: $imili modo $equetur pondus in k |
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1422 grauius e$$e, quàm in I; & v$q; ad S tantùm moueri. & $i rur$us |
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1423 o$tenderent portionem de$cen$us ex S in A, atq; ita deinceps, re |
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1424 ctiorem e$$e æquali de$cen$u ponderis oppo$iti; $emper $equetur |
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1425 libram SI ad AB propius accedere, nunquam tamen in AB per- |
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1426 uenire demon$trabunt. $i igitur libram DE in AB redire demon |
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1427 $trare volunt, nece$$e e$t, vt de$cen$um ponderis ex D in A de di |
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1428 recro capere quantitatem lineæ ex puncto D ip$i AB ad rectos |
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1429 angulos ductæ accipiant. atq; ita, $i æquales de$cen$us DA AN |
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1430 inuicem comparemus, qui æqualiter de directo capient OC CT, |
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1431 cueniet idem pondus in D æquè graue e$$e, vt in A. $i verò por |
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1432 tiones tantum ex D A accipiamus; grauius erit in A, quàm |
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1433 in D. ergo ex diuer$itate tantùm modi con$iderandi, idem pon |
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1434 dus, & grauius, & leuius e$$e continget. non autem exip$a na- |
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1435 <pb n=17> |
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1436 tura rei. In$uper ip$orum $uppo$itio non a$$erit, pondus $ecun |
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1437 dùm $itum grauius e$$e, quantò in eodem $itu minus obliquum |
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1438 e$t principium ip$ius de$cen$us. Suppo$itio igitur $uperius alla |
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1439 ta, hoc e$t, $ecundùm $itum pondus grauius e$$e, quantò in eo |
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1440 dem $itu minus obliquus e$t de$cen$us; non $olum ex his, quæ |
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1441 diximus, vllo modo concedi pote$t; $ed quoniam huius oppo$i |
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1442 tum o$tendere quoq; non e$t difficile: $cilicet idem pondus in |
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1443 æqualibus circumferentiis, quò minus obliquus e$t de$cen$us, ibi |
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1444 minus grauitare. |
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1445 <p>Sint enim vt prius cir |
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1446 cumferentræ AL AM |
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1447 inter $e $e æquales; $itq; |
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1448 punctum L propè F. & |
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1449 connectatur LM, quæ |
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1450 ip$i AB perpendicularis |
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1451 erit. & LX ip$i XM |
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1452 æqualis. deinde propè |
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1453 M inter MG quoduis |
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1454 accipiatur punctum P. |
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1455 fiatq; circumferentia PO |
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1456 circumferentiæ AM æ- |
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1457 qualis. erit punctum O |
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1458 <fig> |
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1459 propè A. connectanturq; CL, CO, CM, CP, OP. & à |
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1460 puncto P ip$i OC perpendicularis ducatur PN. & quoniam cir |
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1461 cumferentia AM circumferentiæ OP e$t æqualis: erit angu- <marg><I>Ex</I> 27 <I>Ter tii.</I></marg> |
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1462 lus ACM æqualis angulo OCP; & angulus CXM rectus re- |
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1463 cto CNP e$t æqualis: erit quoq; reliquus XMC trianguli MCX <marg><I>Ex</I> 32 <I>pri mi.</I></marg> |
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1464 reliquo NPC trianguli PCN æqualis. $ed & latus CM lateri <marg>26 <I>Primi.</I></marg> |
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1465 CP e$t æquale: ergo triangulum MCX triangulo PCN æquale |
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1466 erit. latu$q; MX lateri NP æquale. quare linea PN ip$i LX æqua |
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1467 lis erit. ducatur præterea à puncto O linea OT ip$i AC æqui |
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1468 di$tans, quæ NP $ecet in V. atq; ip$i OT à puncto P perpendi |
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1469 cularis ducatur, quæ quidem inter OV cadere non pote$t; nam |
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1470 cùm angulus ONV $it rectus; erit OVN acutus. quare OVP <marg><I>Ex</I> 13 <I>Pri mi.</I></marg> |
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1471 obtu$us erit. non igitur linea à puncto P ip$i OT intra OV |
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|
1472 <foot>E</foot> |
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1473 <pb> |
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1474 perpendicularis cadet. |
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1475 duo enim anguli vnius |
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1476 trianguli, vnus quidem |
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1477 rectus, alter verò ob- |
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1478 tu$us e$$et. quod e$t im |
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1479 po$sibile. cadet ergo in |
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1480 linea OT in parte VT. |
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1481 $itq; PT. erit PT $ecun |
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1482 dùm ip$os rectus circum |
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1483 ferentiæ OP de$cen$us. |
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1484 Quoniam igitur angulus |
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1485 ONV e$t rectus; erit |
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1486 <marg>19 <I>Primi.</I></marg> linea OV ip$a ON ma |
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1487 ior. quare OT ip$a |
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1488 <fig> |
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1489 quoq; ON maior exi$tet. Cùm itaq; linèa OP angulos $ubten- |
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1490 dat rectos ONP OTP; erit quadratum ex OP quadratis ex |
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1491 <marg>47 <I>Primi.</I></marg> ON NP $imul $umptis æquale. $imiliter quadratis ex OT TP |
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1492 $imulæquale. quare quadrata $imul ex ON NP quadratis ex |
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1493 OT TP $imul æqualia erunt. quadratum autem ex OT maius |
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1494 e$t quadrato ex ON; cum linea OT $it ip$a ON maior. ergo qua |
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1495 dratum ex NP maius erit quadrato ex TP. ac propterea linea |
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1496 TP minor erit linea PN, & linea LX. minus obliquus igitur e$t |
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1497 de$cen$us arcus LA, quàm arcus OP. ergo pondus in L, ex ip |
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1498 $orum dictis, grauius erit, quàm in O. quod ex iis, quæ $upra di |
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1499 ximus e$t manife$tè fal$um, cùm pondus in O grauius $it, quàm |
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1500 in L. non igitur ex rectiori, & obliquiorimotu ita accepto col- |
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1501 ligi pote$t, $ecundùm $itum pondus grauius e$$e, quantò in eo |
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1502 dem $itu minus obliquus e$t de$cen$us. Atq; hinc oritur omnis |
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1503 fermé ip$orum error in hacre, atq; deceptio: nam quamuis per |
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1504 accidens interdum ex fal$is $equatur verum, per $e tamen ex fal |
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1505 $is fal$um $equitur, quemadmodum ex veris $emper verum, nil |
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1506 idcirco mirum, $i dum fal$a accipiunt; illi$q; tanquam veri$si- |
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1507 mis innituntur; fal$i$sima omninò colligunt, atq; concludunt. |
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1508 decipiuntur quinetiam, dùm libræ contemplationem mathemati |
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1509 cè $impliciter a$$ummunt; cùm eius con$ideratio $it pror$us me- |
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1510 chanica: nec vllo modo ab$q; vero motu, ac ponderibus (en- |
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1511 <pb n=18> |
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1512 tibus omninò naturalibus) de ip$a $ermo haberi po$sit: $ine qui- |
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1513 bus eorum, quæ libræ accidunt, veræ caulæ reperiri nullo mo |
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1514 do po$sint. |
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1515 <p>Præterea $i adhuc $up |
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1516 po$itionem conceda- |
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1517 mus; à con$ideratione |
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1518 libræ longè recedunt; |
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1519 dum eo pacto, vt libra |
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1520 DE in AB redire de- |
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1521 beat, di$currunt. $emper |
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1522 enim alterum pondus |
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1523 $eor$um accipiunt, putá |
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1524 D, vel E; ac $i modò vnũ |
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1525 modò alterum in libra |
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1526 con$titutum e$$et, nec |
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1527 vllo modo ambo con- |
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1528 <fig> |
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1529 nexa; cuius tamen oppo$itum omninò fieri oportet; neq; alterum |
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1530 $ine altero rectè con$iderari pote$t; cùm de ip$is in libra con$ti- |
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1531 tutis $ermo habeatur. cùm enim dicunt, de$cen$um ponderis in |
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1532 D minus obliquum e$$e de$cen$u ponderis in E; erit pondus in |
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1533 D per $uppo$itionem grauius pondere in E: quare cùm $it graui- |
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1534 us, nece$$e e$t deor$um moueri, libramq; DE in AB redire: di |
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1535 $cur$us i$te nullius pror$us momenti e$t. Primùm quidem $em- |
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1536 per argumentantur, ac $i pondera in DE de$cendere debeant, |
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1537 vnius tantùm $ine alterius connexione con$iderando de$cen$um. |
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1538 po$tremò tamen ob ponderum de$cen$uum comparationem colli- |
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1539 gentes inferunt, pondus in D deor$um moueri, & pondus in E |
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1540 $ur$um, vtraq; $imul in libra inuicem connexa accipientes. ve- |
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1541 rùm ex ii$demmet, quibus vtuntur, principiis, ac demon$tratio |
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1542 nibus, oppo$itum eius, quod defendere conantur, facillimè col- |
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1543 ligi pote$t. Nam $i comparetur de$cen$us ponderis in D cum a- |
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1544 $cen$u ponderis in E, vt ductis EK DH ip$i AB perpendicula- |
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1545 ribus; cùm angulus DCH $it æqualis angulo ECk; & angulus <marg>15 <I>Primi.</I></marg> |
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1546 DHC rectus æqualis e$t recto E k C; & latus DC lateri CE æqua |
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1547 le: erit triangulum CDH triangulo CEk æquale, & latus DH la- <marg>26 <I>Primi.</I></marg> |
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1548 <foot>E 2</foot> |
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1549 <pb> |
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1550 teri Ek æquale. cùm |
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1551 autem angulus DCA |
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1552 $it angulo ECB æqua- |
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1553 lis: erit quoq; circum- |
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1554 ferentia DA cirferen- |
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1555 tiæ BE æqualis. dum |
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1556 itaq; pondus in D de- |
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1557 $cendit per circumfe- |
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1558 rentiam DA, pondus |
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1559 in E per circumferen- |
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1560 tiam EB ip$i DA æ- |
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1561 qualem a$cendit. & de- |
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1562 $cen$us põderis in D de |
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1563 directo (more ip$orũ) |
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1564 <fig> |
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1565 capiet DH; a$cen$us verò ponderis in E de directo capiet Ek ip |
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1566 $i DH æqualem: erit itaq; de$cen$us ponderis in Da$cen$ui pon |
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1567 deris in E æqualis, & qualis erit propen$io vnius ad motum deor |
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1568 fum, talis etiam erit re$i$tentia alterius ad motum $ur$um. re- |
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1569 $i$tentia $cilicet violentiæ ponderis in E in a$cen$u naturali po- |
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1570 tentiæ ponderis in D in de$cen$u contrà nitendo apponitur; cùm |
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1571 $it ip$i æqualis. quò enim pondus in D naturali potentia deor |
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1572 $um velocius de$cendit, eò tardius pondus in E violenter a$cendit. |
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1573 quare neutrum ip$orum alteri præponderabit, cùm ab æquali non |
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1574 proueniat actio. Non igitur pondus in D pondus in E $ur$um |
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1575 mouebit. $i enim moueret; nece$$e e$$et, pondus in D maiorem |
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1576 habere virtutem de$cendendo, quàm pondus in E a$cendendo; |
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1577 $ed hæc $unt æqualia: ergo pondera manebunt. & grauitas pon- |
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1578 deris in D grauitati ponderis in E æqualis erit. Præterea quoniam |
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1579 $upponunt, quò pondus à linea directionis FG magis di$tat, eò |
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1580 grauius e$$e: Idcirco ductis quoq; à punctis DE ip$i FG perpen |
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1581 dicularibus DO EI; $imili modo demon$trabitur, triangulum |
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1582 CDO triangulo CEI æqualem e$$e: & lineam DO ip$i EI æqua |
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1583 lem. tam igitur di$tat à linea FG pondus in D, quàm pondus in |
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1584 E. ex ip$orum igitur rationibus, atq; $uppo$itionibus, pondera |
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1585 in DE æquè grauia erunt. Amplius quid prohibet, quin libram |
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1586 DE ex nece$sitate in FG moueri $imili ratione o$tendatur? Pri- |
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1587 <pb n=19> |
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1588 mùm quidem ex eorummet demon$trationibus colligi pote$t, a- |
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1589 $cen$um ponderis in E ver$us B rectiorem e$$e a$cen$u ponderis |
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1590 in D ver$us F; hoc e$t minus capere de directo a$cen$um pon- |
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1591 deris in D in arcubus æqualibus a$cen$u ponderis in E. $uppona |
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1592 tur ergo $ecundùm $itum pondus leuius e$$e, quantò in eodem $i- |
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1593 tu minus rectus e$t a$cen$us: quæ quidem $uppo$itio, adeò ma- |
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1594 nife$ta e$$e videtur, veluti ip$orum altera. Quoniam igitur a$cen- |
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1595 $us ponderis in E rectior e$t a$cen$u ponderis in D; per $uppo$i- |
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1596 tionem pondus in D leuius erit pondere in E. ergo pondus in |
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1597 D $ur$um à pondere in E mouebitur, ita vt libra in FG perue |
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1598 niat. atq; ita demon$trari poterit, libram DE in FG moueri. |
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1599 quæ quidem demon$tratio inutilis e$t pror$us, ea$demq; patitur |
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1600 difficultates. licet enim tanquàm verum admittatur pondus in E |
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1601 a$cendendo grauius e$$e pondere in D $imiliter a$cendendo, |
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1602 non tamen ex hoc $equitur, pondus in E de$cendendo grauius |
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1603 e$$e pondere in D a$cendendo. Neutra igitur harum demon- |
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1604 $trationum libram DE, vel in AB redire, vel in FG moue- |
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1605 ri, o$tendentium, vera e$t. |
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1606 <p>Præterea $i ip$orum $uppo$itionem, eorumq; verborum vim |
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1607 rectè perpendamus; alium certè habere $en$um con$piciemus. nam |
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1608 cùm $emper $patium, per quod naturaliter pondus mouetur, à cen |
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1609 tro grauitatis ip$ius ponderis ad centrum mundi, in$tar rectæ li- |
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1610 neæ à centro grauitatis ad centrum mundi productæ, $it $umendum; |
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1611 tantò huiusmodi ponderis de$cen$us, magis, minusuè obliquus |
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1612 dicetur; quantò $ecundùm $patium in$tar prædictæ lincæ de$igna |
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1613 tum, magis, aut minus (naturalem tamen locum petens, $emperq; |
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1614 magis ip$i appropinquans) mouebitur; ita vt tantò obliquior de- |
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1615 $cen$us dicatur, quantò recedit ab eiu$modi $patio: rectiorverò, |
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1616 quantò ad idem accedit. & in hoc $en$u $uppo$itio illa nemini |
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1617 difficultatem parere debet, adeò enim veritas eius con$picua e$t; |
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1618 rationiq; con$entanea: vt nulla pro$us manife$tatione egere vi- |
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1619 deatur. |
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1620 <pb> |
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1621 <p>Si itaq; pondus $olutum in $itu D |
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1622 collocatum ad propium locum mo- |
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1623 ueri debeat; proculdubio po$ito cen- |
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1624 tro mundi S, per lineam DS moue- |
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1625 bitur. $imiliter pondus in E $olutum |
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1626 per lineam ES mouebitur. quare $i |
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1627 (vt rei veritas e$t) ponderis de$cen- |
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1628 $us magis, minu$uè obliquus dicetur |
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1629 $ecundùm rece$$um, & acce$$um ad |
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1630 $patia per lineas DSES de$ignata, |
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1631 iuxtanaturales ip$orum ad propria lo |
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1632 ca lationes; con$picuum e$t, minus |
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1633 obliquum e$$e de$cen$um ip$ius E |
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1634 per EG, quàm ip$ius D per DA: |
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1635 cùm angulum SEG angulo SDA |
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1636 minorem e$$e $upra o$ten$um $it. qua |
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1637 re in E pondus magis grauitabit, |
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1638 quàm in D. quod e$t penitus oppo- |
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1639 $itum eius, quod ip$i o$tendere cona |
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1640 ti $unt. In$urgent autem forta$$e |
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1641 contranos, $i igitur (dicent) pondus |
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1642 in E grauius e$t pondere in D, libra |
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1643 <fig> |
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1644 DE in hoc $itu minimè per$i$tet, quod equid&etilde; tueri propo$uimus: |
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1645 $ed in FG mouebitur. quibus re$pondemus, plurimum referre, $iue |
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1646 con$ideremus pondera, quatenus $unt inuicem di$iuncta, $iue quate |
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1647 nus $unt $ibi inuicem connexa. alia e$t enim ratio ponderis in E $ine |
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1648 connexione ponderis in D, alia verò eiu$dem alteri ponderi con |
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1649 nexi; ita vt alterum $ine altero moueri non po$sit. nam ponde |
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1650 ris in E, quatenus e$t $ine alterius ponderis connexione, rectus |
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1651 naturalis de$cen$us e$t per lineam ES; quatenus verò connexum |
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1652 e$t ponderi in D, eius naturalis de$cen$us non erit amplius per |
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1653 lineam ES, $ed per lineam ip$i CS parallelam. magnitudo enim |
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1654 ex ponderibus ED, & libra DE compo$ita, cuius grauitatis cen- |
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1655 trum e$t C, $i nullibi $u$tineatur, deor$um eo modo, quo reperi |
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1656 tur, $ecundùm grauitatis centrum per rectam à centro grauita |
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1657 tis C ad centrum mundi S ductam naturaliter mouebitur, donec |
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1658 <pb n=20> |
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1659 centrum C in centrum S perueniat. libra igitur DE vná cum pon |
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1660 deribus eo modo, quo reperitur, deor$um mouebitur, ita vt pun- |
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1661 ctum C perlineam CS moueatur, donec C in S, libraq; DE in |
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1662 Hk perueniat; habeatq; libra in Hk eandem, quam prius habe- |
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1663 bat po$itionem; hoc e$t Hk $it ip$i DE æquidi$tans. connect antur |
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1664 igitur DH Ek. manife$tum e$t, dum libra DE in Hk mouetur pun |
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1665 cta DE per lineas DH Ek moueri, quippe exi$tentibus inter $e <marg>33 <I>Prmi.</I></marg> |
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1666 $e, ip$iq; CS æqualibus, & æquidi$tantibus. Quare pondera in |
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1667 DE, quatenus $unt $ibi inuicem connexa, $i ip$orum naturalem mo |
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1668 tum $pectemus, non $ecundùm lineas DS ES, $ed $ecundùm |
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1669 LDH MEk ip$i CS æquidi$tantes mouebuntur. ponderis ve- |
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1670 rò in E liberi, ac $oluti, naturalis propen$io erit per ES: ponderis |
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1671 autem in D $imiliter $oluti erit per DS. ac propterea non e$t incon- |
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1672 ueniens idem pondus modò in E, modò in D, grauius e$$e in E, |
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1673 quàm in D. $i verò pondera in ED $ibi inuicem connexa, quate- |
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1674 nusq; $unt connexa con$iderauerimus; erit ponderis in E natura- |
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1675 lis propen$io per lineam MEK: grauitas enim alterius ponde- |
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1676 ris in D efficit, nè pondus in E per lineam ES grauitet, $ed per |
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1677 Ek. quod ip$um quoq; grauitas ponderis in E efficit, nè $cilicet |
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1678 pondus in D per rectam DS degrauet; $ed $ecundùm DH: vtra- |
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1679 que enim $e impediunt, nè ad propria loca permeent. Cùm igi |
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1680 tur naturalis de$cen$us rectus ponderum in DE $it $ecundùm |
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1681 LDH MEK: erit $imliter rectus eorum a$cen$us $ecundùm ea$ |
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1682 dem lineas HDL KEM. atq; a$cen$us ponderis in E magis, mi |
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1683 nu$uè obliquus dicetur; quantò $ecundùm $patium magis, mi- |
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1684 nu$uè iuxta lineam Mk moucbitur. hocq; pror$us modo iuxta li |
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1685 neam LH $ummendus e$t, tùm de$cen$us, tùm a$cen$us ponde- |
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1686 ris in D. $i itaq; pondus in E deor$um per EG moueretur; pon |
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1687 dus in D $ur$um per DF moueret. & quoniam angulus CEK <marg>29 <I>Primi.</I></marg> |
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1688 æqualis e$t angulo CDL, & angulus CEG angulo CDF æqua- |
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1689 lis; erit reliquus GEK reliquo LDF æqualis. cùm autem $up- |
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1690 po$itio illa, quæ ait, $ecundúm $itum pondus grauius e$$e, quan- |
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1691 tò in eodem $itu minus obliquus e$t de$cen$us; tanquam clara, |
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1692 atq; con$picua admittatur; proculdubio hæc quoq; accipienda |
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1693 erit; nempè, $ecundúm $itum pondus grauius e$$e, quantò in eo- |
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1694 dem $itu minus obliquus e$t a$cen$us. cùm non minus manife$ta, |
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1695 <pb> |
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1696 rationiq; $it con$entanea. æqualis |
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1697 igitur erit de$cen$us ponderis in E |
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1698 a$cen$ui ponderis in D. eandem |
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1699 enim obliquitatem habet de$cen$us |
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1700 ponderis in E, quam habet a$cen- |
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1701 $us ponderis in D; & qualis erit |
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1702 propen$io vnius ad motum deor$um, |
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1703 talis quoq; erit re$i$tentia alterius ad |
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1704 motum $ur$um. nõ ergo pondus in E |
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1705 pondus in D $ur$um mouebit. neq; |
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1706 pondus in D deor$um mouebitur, ita |
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1707 vt $ur$um moueat pondus in E. nam |
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1708 cũ angulus CEB $it ip$i CDA æqua- |
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1709 <marg>29 <I>Primi.</I></marg> lis, & Angulus CEM $it angulo |
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1710 CDH æqualis; erit reliquus MEB |
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1711 reliquo HDA æqualis. de$cen$us |
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1712 igitur ponderis in D a$cen$ui ponde |
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1713 ris in E æqualis erit. non ergo pon |
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1714 dus in D pondus in E $ur$um moue |
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1715 bit. ex quibus $equitur pondera in |
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1716 DE, quatenus $unt $ibi inuicem con |
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1717 nexa, æquè grauia e$$e. |
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1718 <fig> |
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1719 <p>Alia deinde ratio, li- |
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1720 bram $imiliter DE in AB |
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1721 redire o$tendens, cùm in- |
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1722 quiunt, exi$tente trutina in |
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1723 CF meta e$t CG. & quo- |
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1724 niam angulus DCG maior |
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1725 e$t angulo ECG; pondus |
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1726 in D grauius erit pondere |
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1727 in E; ergo libra DE in AB |
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1728 redibit: nihil meo iudicio |
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1729 concludit. figmentumq; |
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1730 hoc de trutina, & meta po- |
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1731 tius omittendum, ac $ilen- |
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1732 <fig> |
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1733 <pb n=21> |
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1734 tio prætereundũ e$$et, quàm verbũ vllũ in eius confutatione $umen |
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1735 dum; cùm $it pror$us voluntarium. nece$sitas enim cur pondus |
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1736 in D ex maiore angulo $it grauius; curq; maior angulus maioris |
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1737 $it cau$a grauitatis; nu$quam apparet. $i autem comparentur in- |
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1738 uicem anguli, cùm angulus GCD $it æqualis angulo FCE; $i angu |
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1739 lus GCD e$t cau$a grauitatis; quare angulus FCE $imiliter gra- |
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1740 uitatis non e$t cau$a? Huius autem rei eam in medium rationem |
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1741 afferre videntur, quoniam CG e$t meta, & CF trutina. $i (inquiunt) |
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1742 CG e$$et trutina, & CF meta, tunc angulus FCE grauitatis e$$et |
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1743 cau$a; non autem DCG ip$i æqualis. quæ quidem ratio imma- |
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1744 ginaria pror$us, ac voluntaria e$$e videtur. quid enim refert, $iue tru |
|
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1745 tina $it in CF, $iue in CG, cùm libra DE in eodem $emper pun- |
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1746 cto C $u$tineatur? Vt autem eorum deceptio clarius appa- |
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1747 reat. |
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1748 <p>Sit eadem libra AB, cu- |
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1749 ius medium C. $it deinde |
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1750 tota FG trutina. eaq; im |
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1751 mobilis exi$tat; quælibram |
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1752 AB in puncto C $u$tineat. |
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1753 moueaturq; libra in DE. & |
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1754 quoniam trutina e$t, & $u- |
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1755 pra, & infra libram, quis |
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1756 nam angulus erit cau$a gra- |
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1757 uitatis, cùm libra DE in |
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1758 <fig> |
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1759 eod&etilde; $emper puncto $u$tineatur? dicent for$an, $i trutina à potentia |
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1760 in F $u$titencatur, tunc CG erit tanquam meta, & angulus |
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1761 DCG grauitatis erit cau$a. $i verò $u$tineatur in G, tunc FCE |
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1762 erit cau$a grauitatis, CF verò tanquam meta erit. cuius quidem |
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1763 rei nulla videtur e$$e cau$a, ni$i immaginaria. meta enim (quod |
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1764 aiunt) nullam pror$us vim attractiuam, quandoq; ex maioris an- |
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1765 guli parte, quandoq; ex parte minoris habere videtur. Verùm à dua |
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1766 bus potentiis $u$tineatur trutina, in F $cilicet, & in G, quod præ ne |
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1767 ce$sitate fieri pote$t, veluti $i potentia in F $it adeò debilis, vt ex $e |
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1768 ip$a medietatem tantùm ponderis $u$tinere quæat: $itq; potentia in |
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1769 Gip$i potentiæ in F æqualis, vtræq; aut&etilde; $imul libram vná cum pon |
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1770 deribus $u$tineant. tunc quis nam angulus erit cau$a grauitatis? non |
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1771 <foot>F</foot> |
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1772 <pb> |
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1773 FCE, quia trutina e$t in |
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1774 CF, & in F $u$tinetur. neq; |
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1775 DCG, cùm trutina $it in |
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1776 CG, & in G quoq; $u$ti |
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1777 neatur; non igitur anguli |
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1778 grauitatis cau$a erunt. ergo |
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1779 neq; libra DE ab hoc $itu |
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1780 ob hanc cau$am mo uebi- |
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1781 <marg><I>Cardanus.</I></marg> tur. Hanc autem eorum |
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1782 $ententiam dupliciter con- |
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1783 <fig> |
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1784 firmare videntur. primùm quidem a$$erunt Ari$totelem in quæ$tio |
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1785 nibus mechanicis has duas tantùm quæ$tiones propo$ui$$e; eiu$q; |
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1786 demon$trationes, tum maiori, & minori angulo, tùm trutinæ po$i |
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1787 tioni inniti. Af$irmant deinde experientiam hoc idem docere; |
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1788 hoc e$t libram DE trutina exi$tente in CF, in AB horizonti |
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1789 æquidi$tantem redire. quando autem trutina e$t in CG, in FG |
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1790 moueri. Verùm neq; Ari$toteles, neq; experientia huic eorum |
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1791 opinioni fauent, quin potius aduer$antur. quantùm enim atti- |
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1792 net ad experientiam decipiuntur, ip$a quidem experientia ma- |
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1793 nife$tum e$t hoc accidere, quando libræ quoq; centrum, vel $u- |
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1794 pra, vel infra libram fuerit collocatum: non autem trutina dun |
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1795 taxat $upra, vel infra exi$tente, id contingere. |
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1796 <pb n=22> |
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1797 <p>Nam $i libra AB habeat |
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1798 centrum C $upra libram; |
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1799 $itq; trutina CD infra li- |
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1800 bram; moueaturq; libra in |
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1801 EF; tunc EF rur$us in AB |
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1802 horizonti æquidi$tantem <marg>2 <I>Huius.</I></marg> |
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1803 redibit. $imiliter $i libra |
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1804 centrum C habeat infra li |
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1805 bram, $itq; trutina CD $u |
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1806 pra libram, & moueatur |
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1807 libra in EF; patet libram <marg>3 <I>Huius.</I></marg> |
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1808 ex parte F deor$um moue |
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1809 ri, trutina $upra libram e- |
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1810 xi$tente. & in quocunq; a- |
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1811 lio $itu fuerit trutina, idem |
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1812 $emper eueniet. non igitur |
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1813 trutina, $ed centrum libræ |
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1814 harum diuer$itatum cau- |
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1815 $a erit. |
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1816 <fig> |
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1817 <p>Animaduertendum e$t |
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1818 itaq; in hac parte difficulter materialem libram con$titui po$$e, |
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1819 quæ in vno tantùm puncto $u$tineatur; quemadmodum mente |
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1820 concipimus. brachiaq; ab eiu$modi centro adeò æqualia habeat, |
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1821 non $olum in longitudine, verùm etiam in latitudine, & profun |
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1822 ditate, vt omnes partes hinc indé ad vnguem æqueponderent. |
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1823 hoc enim materia difficilimè patitur. quocirca $i centrum in ip$a |
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1824 libra e$$e con$iderauerimus, ad $en$um confugiendum non e$t: |
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1825 cùm artificilia ad $ummum illud perfectionis gradum ab artifice |
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1826 deduci minimè po$sint. In aliis verò experientia quidem appa- |
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1827 rentia docere poterit; proptereaquod, quamquam centrum libræ |
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1828 $it $emper punctum, quando tamen $upra libram fuerit, parùm re- |
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1829 fert, $i libra in eo puncto adamu$$im minimè $u$tineatur; quia cùm |
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1830 $it $emper $upra libram, idem $emper eueniet. $imili quoq; modo |
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1831 quando e$t infra libram: quod tamen non accidit centro in ip$a li- |
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1832 bra exi$tente. $i enim ad vnguem $emper in illo medio non $u- |
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1833 $tineatur, diuer$itatem efficiet; cùm facillimum $it, centrum il- |
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1834 <foot>F 2</foot> |
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1835 <pb> |
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1836 lud, dùm libra mouetur, proprium mutare $itum. |
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1837 <p>Quòd autem Ari$toteles duas tantùm quæ$tiones propo- |
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1838 $uerit, cur $cilicet trutina $uperius exi$tente, $i libra non $it |
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1839 horizonti æquidi$tans in æquilibrium, hoc e$t horizonti æqui |
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1840 di$tans redit: $i autem trutina deor$um fuer it con$tituta, non |
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1841 redit; $ed adhuc $ecundùm partem depre$$am mouetur: verum |
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1842 quidem e$t. non tamen eius demon$trationes maiori, & mino |
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1843 ri angulo, po$itioniqué trutinæ (vt ip$i dicunt) innituntur. In |
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1844 hoc enim mentem philo$ophi a$ignantis rationem diuer$itatis |
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1845 motuum libræ minimè attingunt. tantùm enim abe$t philo$o- |
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1846 phum has diuer$itates in angulos referre, vt potius in cau$a e$$e |
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1847 dicat magnitudinis alterius brachii libræ exce$$um à perpendiculo, |
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1848 modò ex vna, modò ex altera parte contingentem. |
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1849 <p>Vt trutina $uperius in |
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1850 CF exi$tente, perpendicu |
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1851 lum erit FCG, quod $e- |
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1852 cundùm ip$um in centrum |
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1853 mundi $emper vergit; |
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1854 quod quidem libram mo- |
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1855 tam in DE in partes di- |
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1856 uidit inæquales; & maior |
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1857 pars e$t ver$us D: id au- |
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1858 tem, quod plus e$t, deor |
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1859 $um fertur; ergo ex par- |
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1860 te D deor$um libra moue |
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1861 bitur, donec in AB re- |
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1862 deat. $i verò trutina $it |
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1863 <fig> |
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1864 in CG deor$um, erit GCF perpendiculum, quod libram DE |
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1865 in partes inæquales $imiliter diuidit: maior autem pars erit ver$us |
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1866 E; quare ex parte E deor$um libra mouebitur. quod vt rectè in- |
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1867 telligatur, cùm trutina e$t $upra libram, libræ quoq; centrum $u- |
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1868 pra libram e$$e intelligendum e$t; & $i deor$um, centrum quoque |
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1869 deor$um: vt infra patebit. Aliter ip$a Ari$totelis demon$tratio |
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1870 nihil concluderet. exi$tente enim centro in ip$a libra, vt in C; quo- |
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1871 cunq; modo moueatur libra, nunquam perpendiculum FG libram, |
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1872 <pb n=23> |
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1873 ni$i in puncto C, & in partes diuidet æquales. quare Ari$totelis |
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1874 $ententia ip$is non $olum non fauet, verùm etiam maximè aduer- |
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1875 $atur. quòd non $olum ex $ecunda, & rertia huius liquet; verùm |
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1876 quia exi$tente centro $upra libram pondus eleuatum maiorem |
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1877 propter $itum acquirit grauitatem. ex quò contingit redditus li- |
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1878 bræ ad æqualem horizonti di$tantiam. è contra verò, quando |
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1879 centrum e$t infra libram. Quæ omnia hoc modo o$tendentur; |
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1880 $upponendo ea, quæ $upra declarata $unt. $cilicet pondus ex quò |
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1881 loco rectius de$cendit, grauius fieri. & ex quo rectius a$cendit, gra |
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1882 uius quoq; reddi. |
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1883 <p>Sit libra AB horizonti |
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1884 æquidi$tans, cuius centrum |
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1885 C $it $upra libram, perpen- |
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1886 diculumq; $it CD. $intq; in |
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1887 AB ponderum æqualium |
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1888 centra grauit atis po$ita: mo |
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1889 taq; $it libra in EF. Dico |
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1890 pondus in E maiorem ha- |
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1891 bere grauitatem, quàm pon |
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1892 dus in F. & ob id libram |
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1893 EF in AB redire. Produ |
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1894 catur primùm CD v$q; ad |
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1895 mundi centrũ, quod $it S. de |
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1896 inde AC CB EC CF HS |
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1897 cõnectantur, à puncti$q; EF |
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1898 ip$i HS æquidi$tantes du |
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1899 cantur Ek GFL. Quoniam |
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1900 igitur naturalis de$cen$us re |
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1901 ctus totius magnitudinis, |
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1902 libræ $cilicet EF $ic con$ti- |
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1903 tutæ vná cum ponderibus, |
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1904 e$t $cundùm grauitatis cen |
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1905 trum H per rectam HS; erit |
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1906 <fig> |
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1907 quoq; ponderum in EF ita po$sitorum de$cen$us $ecundùm re- |
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1908 ctas Ek FL ip$i HS parallelas; $icuti $upra demon$trauimus. |
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1909 <pb> |
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1910 De$cen$us igitur, & a$cen- |
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1911 $us ponderum in EF ma- |
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1912 gis, minu$uè obliquus di- |
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1913 cetur $ecundùm acce$$um, |
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1914 & rece$$um iuxta lineas Ek |
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1915 FL de$ignatum. Quoniã au |
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1916 t&etilde; duo latera AD DC duo |
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1917 bus lateribus BD DE $unt |
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1918 æqualia; anguliq; ad D $unt |
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1919 <marg>4 <I>Primi.</I></marg> recti; erit latus AC lateri |
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1920 CB æquale. & cùm pun- |
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1921 ctum C $it immobile; dum |
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1922 puncta AB mouentur, cir |
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1923 culi circumferentiam de$cri |
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1924 bent, cuius $emidiameter |
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1925 erit AC. quare centro C, |
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1926 circulus de$cribatur AEBF. |
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1927 puncta AB EF in circuli |
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1928 circumferentia erunt. $ed |
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1929 cùm EF $it ip$i AB æqua |
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1930 <marg><I>Ex</I> 28 <I>Ter tii.</I></marg> lis; erit circumferentia |
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1931 EAF circumferentiæ AFB |
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1932 æqualis. quare dempta |
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1933 <fig> |
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1934 communi AF, erit circumferentia EA circumferentiæ FB æqua |
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1935 lis. Quoniam autem mixtus angulus CEA e$t æqualis mixto |
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1936 CFB; & HFB ip$o CFB e$t maior; angulus verò HEA ip$o |
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1937 CEA minor; erit angulus HFB angulo HEA maior. à quibus |
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1938 <marg>29 <I>Primi.</I></marg> $i auferantur anguli HFG HEk æquales; erit angulus GFB an |
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1939 gulo kEA maior. ergo de$cen$us ponderis in E minus obliquus |
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1940 erit a$cen$u ponderis in F. & quamquam pondus in E de$cen |
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1941 dendo, & pondus in F a$cendendo per circumferentias mouean |
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1942 tur æquales; quia tamen pondus in E ex hoc loco rectius de$cen |
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1943 dit, quàm pondus in F a$cendit: idcirco naturalis potentia pon |
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1944 deris in E re$i$tentiam violentiæ ponderis F $uperabit. quare |
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1945 maiorem grauitatem habebit pondus in E, quàm pondus in F. |
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1946 ergo pondus in E deor$um, pondus verò in F $ur$um mouebitur: |
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1947 <pb n=24> |
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1948 donec libra EF in AB redeat. quod demon$trare oportebat. |
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1949 <p>Huius autem effectus ratio ab Ari$totele po$ita, hic manife$ta in <marg><I>Ari$totelis ratio.</I></marg> |
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1950 tueri pote$t. $it enim punctum N vbi CS EF $e inuicem $ecant. |
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1951 & quoniam HE e$t ip$i HF æqualis; erit NE maior NF. li- |
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1952 nea ergo CS, quam perpendiculum vocat, libram EF in partes di |
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1953 uidet inæquales. cùm itaq; pars libræ NE $it maior NF; atq; id, |
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1954 quod plus e$t, nece$$e e$t, deor$um ferri: libra ergo EF ex parte E |
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1955 deor$um mouebitur, donec in AB redeat. |
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1956 <p>Ex iis præterea, quæ ha |
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1957 ctenus dicta $unt inferre li |
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1958 cet, libram EF velocius ab |
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1959 eo $itu in AB moueri; vndè |
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1960 linea EF in directum pro- |
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1961 tracta in centrum mundi |
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1962 perueniat. vt $it EFS recta |
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1963 linea. & quoniam CD |
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1964 CH, $unt inter $e $e æqua |
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1965 les. $i igitur centro C, $pa |
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1966 tioq; CD, circulus de$cri- |
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1967 batur DHM; erunt pun- |
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1968 cta DH in circuli circum- |
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1969 ferentia. Quoniam au- |
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1970 tem CH ip$i EF e$t per- |
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1971 pendicularis; continget li- |
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1972 nea EHS circulum DHM |
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1973 in puncto H. pondus igi- |
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1974 tur in H ($icuti $upra de- |
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1975 mon$trauimus) grauius |
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1976 <fig> |
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1977 erit, quàm in alio $itu circuli DHM. ergo magnitudo ex EF |
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1978 ponderibus, & libra EF compo$ita, cuius centrum grauitatis e$t |
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1979 in H, in hoc $itu magis grauitabit, quàm in quocunq; alio $itu |
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1980 <pb> |
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1981 circuli fuerit punctum H. |
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1982 ab hoc igitur $itu velo- |
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1983 cius, quàm à quocunq; |
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1984 alio mouebitur. & $i H |
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1985 propius fuerit ip$i D mi |
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1986 nus grauitabit, minu$q; |
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1987 ab eo $itu mouebitur. |
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1988 $emper enim de$cen$us |
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1989 obliquior e$t, & minus re |
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1990 ctus. libra ergo EF velo |
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1991 cius ab hoc $itu mouebi- |
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1992 tur, quàm ab alio $itu. & |
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1993 $i propius ad AB acce- |
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1994 det, inde minus mouebi |
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1995 tur. Deinde quò longius |
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1996 punctum H à puncto C |
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1997 di$tabit, velocius moue- |
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1998 bitur; quod nõ $olũ ex Ari |
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1999 $totele in principio quæ$t- |
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2000 io num mechanicarum, & |
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2001 <fig> |
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2002 ex $uperius dictis patet; verùm etiam ex iis, quæ infra in $exta |
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2003 propo$itione dicemus, manife$tum erit. libra igitur EF, quò ma |
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2004 gis ab eius centro di$tabit, adhuc velocius mouebitur. |
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2005 <pb n=25> |
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2006 <p>Sit deinde libra AB, |
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2007 cuius centrum C $it infra li |
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2008 bram; $intq; in AB pon |
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2009 deraæqualia; libraq; $it |
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2010 mota in EF. Dico maio- |
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2011 rem habere grauitatem |
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2012 pondus in F, quàm pondus |
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2013 in E. atq; ideo libram EF |
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2014 deor$um ex parte F moue- |
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2015 ri. Producatur DC ex |
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2016 vtraq; parte v$q; ad mun- |
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2017 di centrum S, & v$q; ad |
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2018 O, lineaq; HS ducatur, |
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2019 cui à punctis EF æquidi- |
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2020 $tantes ducantur GEk FL; |
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2021 connectanturq; CE CF: |
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2022 atq; centro C, $patioq; CE |
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2023 circulus de$cribatur AEO |
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2024 BF. $imiliter demon$tra- |
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2025 bitur puncta ABEF in |
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2026 circuli circumferentia e$$e; |
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2027 de$cen$umq; libræ EF vná |
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2028 cum ponderibus rectum $e |
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2029 cundùm lineam HS fieri; |
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2030 ponderumq; in EF $ecun |
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2031 <fig> |
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2032 dùm lineas GK FL ip$i HS æquidi$tantes. Quoniam autem an |
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2033 gulus CFP æqualis e$t angulo CEO: erit angulus HFP angulo |
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2034 HEO maior. angulus verò HFL æqualis e$t angulo HEG. à <marg>29 <I>Primi.</I></marg> |
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2035 quibus igitur $i demantur anguli HFP HEO, erit angulus |
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2036 LFP angulo GEO minor. quare de$cen$us ponderis in F rectior |
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2037 erit a$cen$u ponderis in E. ergo naturalis potentia ponderis in |
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2038 F re$i$tentiam violentiæ ponderis in E $uperabit. & ideo ma- |
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2039 iorem habebit grauitatem pondus in F, quàm pondus in E. |
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2040 Pondus igitur in F deor$um, pondus verò in E $ur$um mo- |
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2041 uebitur. |
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2042 <p>Ari$totelis quoq; ratio hic per$picua erit. $it enim punctum <marg><I>Ari$totelis ratio.</I></marg> |
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2043 <foot>G</foot> |
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2044 <pb> |
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2045 N vbi CO EF $e inuicem |
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2046 $ecant; erit NF maior |
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2047 NE. & quoniam CO per |
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2048 pendiculum ($ecundùm |
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2049 ip$um) libram EF in par |
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2050 tes inæquales diuidit, & |
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2051 maior pars e$t ver$us F, hoc |
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2052 e$t NF; libra EF ex par |
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2053 te F deor$um mouebitur: |
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2054 cùmid, quod plus e$t, deor |
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2055 $um feratur. |
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2056 <p>Similiter, éx dictis |
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2057 quoq; eliciemus libram EF |
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2058 centrum habens infra li- |
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2059 bram, quò magis à $itu |
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2060 AB di$tabit, velocius mo |
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2061 ueri. centrum enim graui |
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2062 tatis H, quò magis á pun- |
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2063 cto D di$tat, eò volecius |
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2064 pondus ex EF ponderibus, |
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2065 libraq; EF compo$itum |
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2066 mouebitur, donec angulus |
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2067 CHS rectus euadat. ad- |
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2068 huc in$uper velocius moue |
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2069 bitur, quò libram à centro |
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2070 C magis di$tabit. |
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|
2071 <fig> |
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2072 <p>Ex ip$orum quinetiam rationibus, ac fal$is $upo$itionibus iam |
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2073 declaratos libræ effectus, ac motus deducere, ac manife$tare libet; |
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2074 vt quanta $it veritatis efficacia appareat, quippè ex fal$is etiam |
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2075 eluce$cere contendit. |
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|
2076 <pb n=26> |
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2077 <p>Exponantur eadem, $ci |
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2078 licet $it circulus AEBF; |
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2079 libraqué AB, cuius cen- |
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2080 trum C $it $upra libram, |
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2081 moueatur in EF. dico |
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2082 pondus in E maiorem ibi |
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2083 habere grauitatem, quàm |
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2084 pondus in F; libramq; EF |
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|
2085 in AB redire. Ducantur |
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|
2086 à punctis EF ip$i AB |
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|
2087 perpendiculares EL FM, |
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2088 quæ inter $e æquidi$tan- <marg>28 <I>Primi.</I></marg> |
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2089 <fig> |
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2090 tes erunt; $itq; punctum N, vbi AB EF $e inuicem $ecant. |
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2091 Quoniam igitur angulus FNM e$t æqualis angulo ENL, & an- <marg>15 <I>Primi.</I></marg> |
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2092 gulus F MN rectus recto ELN æqualis, ac reliquus NFM reli- <marg>29 <I>Primi.</I></marg> |
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2093 quo NEL e$t etiam æqualis; erit triangulum NLE triangu |
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2094 lo NMF $imile. vt igitur NE ad EL, ita NF ad FM; & per <marg>4 <I>Sexti.</I></marg> |
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2095 mutando vt EN ad NF, ita EL ad FM. $ed cùm $it HE ip$i <marg>16 <I>Quinti.</I></marg> |
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2096 HF æqualis, erit EN maior NF; quare & EL maior erit FM. |
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2097 & quoniam dum pondus in E per circumferentiiam EA de$cendit, |
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2098 pondus in F per circumferentiam FB ip$i circumferentiæ EA |
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2099 æqualem a$cendit; de$cen$u$q; ponderis in E de directo (vt ip- |
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2100 $i dicunt) capit EL: a$cen$us verò ponderis in F de directo ca- |
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2101 pit FM; minus de directo capiet a$cen$us ponderis in F, quàm |
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2102 de$cen$us ponderis in E. maiorem igitur grauitatem habebit pon |
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2103 dus in E, quàm pondus in F. |
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2104 <p>Producatur CD ex vtraq; parte in OP, quæ lineam EF in |
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2105 puncto S $ecet. & quoniam (vt aiunt) quò magis pondus à li- |
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2106 nea directionis OP di$tat, eò fit grauius; idcirco hoc quoq; me |
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2107 dio pondus in E maiorem habere grauitauitatem pondere in F o- |
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2108 $tendetur. Ducantur à punctis EF ip$i OP perpendiculares EQ |
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2109 FR. $imiliratione o$tendetur, triangulum QES triangulo RFS |
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2110 $imile e$$e; lineamq; EQ ip$a RF maiorem e$$e. pondus itaq; |
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2111 in E magis à linea OP di$tabit, quàm pondus in F; ac propterea |
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2112 pondus in E maiorem habebit grauitatem pondere in F. ex quibus |
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2113 reditus libræ EF in AB manife$tus apparet. |
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2114 <foot>G 2</foot> |
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2115 <pb> |
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2116 <p>Si autem centrum libræ |
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2117 $it infra libram, tunc pon- |
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2118 dus depre$$um maiorem |
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2119 habere grauitatem eleuato |
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2120 ii$dem mediis o$tendetur. |
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2121 ducantur à punctis EF ip- |
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2122 $i AB perpendiculares EL |
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2123 FM. $imiliter demon$tra |
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2124 bitur EL maiorem e$$e |
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2125 FM; & ob id de$cen$us |
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2126 ponderis in F minus de di |
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2127 recto capiet, quàm a$cen- |
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2128 <fig> |
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2129 $us ponderis in E: quocirca re$i$tentia violentiæ ponderis in E $u |
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2130 perabit naturalem propen$ionem ponderis in F. ergo pondus in E |
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2131 pondere in F grauius erit. |
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2132 <p>Producatur etiam CD ex vtraq; parte in OP; ip$iq; à punctis |
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2133 EF perpendiculares ducantur EQ FR. eodem pror$us modo |
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2134 o$tendetur, lineam EQ maiorem e$$e FR. pondus ideò in E ma |
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2135 gis à linea directionis OP di$tabit, quàm pondus in F. maio- |
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2136 rem igitur grauitatem habebit pondus in E, quàm pondus in F. |
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2137 ex quibus $equitur, libram EF ex parte E deor$um moueri. |
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2138 <p>Ari$toteles itaq; has duas tantùm quæ$tiones propo$uit, ter- |
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2139 tiamq; reliquit; $cilicet cùm centrum libræ in ip$a e$t libra: hanc |
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2140 autem ommi$sit, vt notam, quemadmodum res valde notas præ- |
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2141 termittere $olet. nam cui dubium, $i pondus in eius centro gra |
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2142 uitatis $u$tineatur, quin maneat? Ea verò, quæ ex ip$ius $enten |
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2143 tia attulimus, aliquis reprehendere po$$et, nos integram eius $enten |
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2144 tiam minimè protuli$$e affimans. nam cùm in $ecunda parte $e |
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2145 cundæ quæ$tionis proponit, cur libra, trutina deor$um con$tituta, |
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2146 quando deor$um lato pondere qui$piam id amouet, non a$cen |
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2147 dit, $ed manet? non a$$erit adhuc libram deor$um moueri; $ed |
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2148 manere. quod in vltima quoq; conclu$ione colligi$$e videtur. Ve |
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2149 rùm hoc non $olum nobis non repugnat, $ed $i rectè intelligitur, |
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2150 maximè $uffragatur. |
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2151 <pb n=27> |
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2152 <p>Sit enim libra AB |
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2153 horizonti æquidi$tans, |
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2154 cuius centrum E $it |
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2155 infra libram. quia ve |
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2156 rò Ari$toteles libram, |
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2157 $icuti actu e$t, con$ide |
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2158 rat; ideò nece$$e e$t |
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2159 trutinam, vel aliquid |
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2160 aliud infra centrum E |
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2161 collocare, vt EF |
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2162 (quod quidem truti- |
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2163 na erit) ita vt centrum |
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2164 E fu$tineat. $itq; per- |
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2165 <fig> |
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2166 pendiculum ECD. & vt libra AB ab hoc moueatur $itu; dicit |
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2167 Ari$toteles, ponatur pondus in B, quod cùm $it graue, libram ex |
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2168 parte B deor$um mouebit; putá in G. ita vt propter impedimen |
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2169 tum deor$um amplius mouerinon poterit. non enim dicit Ari |
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2170 $toteles, moueatur libra ex parte B deor$um, quou$q; libuerit; dein |
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2171 de relinquatur, vt nos diximus: $ed præcipit, vt in ip$o B po- |
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2172 natur pondus, quod ex ip$ius natura deor$um $emper mouebi- |
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2173 tur; donec libra trutinæ, $iue alicui alii adhæreat. & quando B erit |
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2174 in G, erit libra in GH; in quo $itu, ablato pondere, manebit: |
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2175 cùm maior pars libræ à perpendiculo $it ver$us G, quæ e$t DG, |
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2176 quàm DH. nec deor$um amplius mouebitur; nam libra, vel |
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2177 trutinæ, vel alteri cuipiam, quod centrum libræ $u$tineat, incum |
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2178 bet. $i enim huic non adhæreret, libra ex parte G deor$um ex |
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2179 ip$ius $ententia moueretur; cùm id, quod plus e$t, $cilicet DG, |
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2180 deor$um ferri $it nece$$e. |
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2181 <p>Cæterum quis adhuc dicere poterit, $i paruum imponatur pon |
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2182 dus in B, mouebitur quidem libra deor$um, non autem v$q; ad |
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2183 G. in quò $itu $ecundùm Ari$totelem, ablato pondere, mane- |
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2184 re deberet. quod experimento patet; cùm in vna tantùm libræ |
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2185 extremitate, impo$ito onere, hocq; vel maiore, vel minore, libra |
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2186 plus, minu$uè inclinetur. Quod e$t quidem veri$$imum, centro $upra |
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2187 libram, non autem infra, neq; in ip$a libra collocato. Vt exempli |
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2188 gratia. |
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2189 <pb> |
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2190 <p>Sit libra horizonti æ- |
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2191 quidi$tans AB, cuius cen |
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2192 trum C $it $upra libram, |
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2193 perpendiculumq; CD ho |
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2194 rizonti perpendiculare, |
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2195 quod ex parte D produca |
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2196 tur in H. Quoniam enim |
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2197 con$iderata libræ grauita- |
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2198 te, erit punctum D libræ |
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2199 centrum grauitatis. $iergo |
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2200 in B paruum imponatur |
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2201 pondus, cuius centrum |
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2202 <fig> |
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2203 grauitatis $it in puncto B; magnitudinis ex libra AB, & pondere |
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2204 in B compo$itæ non erit amplius centrum grauitatis D; $ed erit in |
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2205 <marg>6 <I>Primi Ar chim. de æquep.</I></marg> linea DB, vt in E: ita vt DE ad EB $it, vt pondus in B ad gra- |
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2206 uitatem libræ AB. Connectatur CE. Quoniam autem pun- |
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2207 ctum Ce$t immobile, dum libra mouetur, punctum E circuli cir |
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2208 cumferentiam EFG de$cribet, cuius $emidiameter CE, & cen- |
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2209 trum C. quia verò CD horizonti e$t perpendicularis, linea CE |
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2210 horizonti perpendicularis nequaquam erit. quare magnitudo ex |
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2211 AB, & pondere in B compo$ita minimè in hoc $itu manebit; $ed |
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2212 <marg>1. <I>Huius.</I></marg> deor$um $ecundùm eius grauitatis centrum E per circumferen- |
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2213 tiam EFG mouebitur; donec CE horizonti perpendicularis eua |
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2214 dat; hoc e$t, donec CE in CDF perueniat. atq; tunc libra AB |
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2215 mota erit in kL, in quo $itu libra vná cum pondere manebit. nec |
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2216 deor$um amplius mouebitur. Si verò in B ponatur pondus graui- |
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2217 us; centrum grauitatis totius magnitudinis erit ip$i B propius, vt in |
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2218 M. & tunclibra deor$um, doneciuncta CM in linea CDH per |
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2219 ueniat, mouebitur. Ex maiore igitur, & minore pondere in B po |
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2220 $ito, libra plus, minu$uè inclinabitur. ex quo $equitur pondus B |
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2221 quarta circuli parte minorem $emper circumferentiam de$cribe- |
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2222 re, cùm angulus FCE $it $emper acutus. nunquam enim punctum |
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2223 B v$q; ad lineam CH perueniet, cùm centrum grauitatis ponde- |
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2224 ris, & libræ $imul $emper inter DB exi$tat. quò tamen pondus |
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2225 in B grauius fuerit, maiorem quoq; circumferentiam de$cribet. |
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2226 eò enim magis punctum B ad lineam CH accedet. |
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2227 <pb n=28> |
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2228 <p>Habeat autem libra AB |
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2229 centrum C in ip$a libra, atq; |
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2230 in eius medio: erit C libræ |
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2231 centrum quoq; grauitatis; |
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2232 à quo ip$i AB, horizontiq; |
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2233 perpendicularis ducatur FC |
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2234 G. ponatur deinde in B |
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2235 quoduis pondus; erit totius |
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2236 magnitudinis centrum gra- |
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2237 uitatis putá in E; ita vt CE |
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2238 <fig> |
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2239 ad EB $it, vt pondus in B ad libræ grauitatem. & quoniam CE |
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2240 non e$t horizonti perpendicularis, libra AB, atq; pondus in B |
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2241 in hoc $itu nunquam manebunt; $ed deor$um ex parte B mouebun |
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2242 tur, donec CE horizontifiat perpendicularis. hoc e$t donec li- |
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2243 bra AB in FG perueniat. ex quo patet, quolibet pondus in B |
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2244 circuli quartam $emper de$cribere. |
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2245 <p>Sit autem centrum Cin- |
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2246 fra libram AB. $itq; DCE |
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2247 perpendiculum. $imiliter |
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2248 po$ito in B pondere, cen- |
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2249 trum grauitatis magnitudi |
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2250 nis ex AB libra, & ponde |
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2251 re in B compo$itæ in linea |
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2252 DB erit; vt in F; ita vt DF |
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2253 ad FB $it, vt pondus in B |
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2254 <fig> |
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2255 ad libræ pondus. Iungatur CF. & quoniam CD horizonti e$t |
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2256 perpendicularis; linea CF horizonti nequaquam perpendicula- |
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2257 ris exi$tet. quare magnitudo ex AB libra, ac pondere in B com |
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2258 po$ita in hoc $itu nunquam per$i$tet; $ed deor$um, ni$i aliquid |
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2259 impediat, mouebitur; donec CF in DCE perueniat: in quo $itu |
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2260 libra vná cum pondere manebit. & punctum B erit vt in G, atq; |
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2261 punctum A in H, libraq; GH non amplius centruminfra, $ed $u |
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2262 pra ip$am habebit. quod idem $emper eueniet; quamurs mini- |
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2263 mum imponatur pondus in B. ergo priu$quam B perueniat ad |
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2264 G; nece$$e e$t libram, $iue trutinæ deor$um po$itæ, vel alicui |
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2265 <pb> |
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2266 alteri, quod centrum C $u- |
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2267 $tineat, occurrere; ibiq; ad- |
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2268 hærere. ex hoc $equitur, pon |
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2269 dus in B vltra lineam Dk |
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2270 $emper moueri; ac circuli |
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2271 quarta maiorem $emper cir |
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2272 cumfer&etilde;tiam de$cribere: e$t |
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2273 enim angulus FCE $emper |
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2274 obtu$us, cùm angulus DCF |
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2275 $emper $it acutus. quò au- |
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2276 <fig> |
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2277 tem pondus in B fuerit leuius, maiorem tamen adhuc circumfe- |
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2278 rentiam de$cribet. nam quò pondus in G leuius fuerit, eò ma- |
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2279 gis pondus in G eleuabitur; libraq; GH ad $itum horizontiæqui |
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2280 di$tantem propius accedet. quæ omnia ex iis, quæ $upra dixi- |
|
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2281 mus, manife$ta $unt. |
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2282 <p>His demon$tratis. Manife$tum e$t, centrum libræ cau$am e$$e |
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2283 diuer$itatis effectuum in libra. atq; patet omnes Archimedis de |
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2284 æqueponderantibus propo$itiones ad hoc pertinentes in omni $itu |
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2285 veras e$$e. hoc e$t $iue libra $it horizonti æquidi$tans, $iue non: |
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2286 dummodo centrum libræ in ip$a $it libra; quemadmodum ip$e |
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2287 con$iderat. & quamquam libra brachia habeat inæqualia, idem eue |
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2288 niet; eodemq; pro$us modo o$tendetur, centrum libræ diuer$imo |
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2289 dè collocatum varios producere effectus. |
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2290 <p>Sit enim libra AB hori- |
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2291 zonti æquidi$tans; & in AB |
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2292 $int pondera inæqualia, quo |
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2293 rum grauitatis centrum $it |
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2294 C: $u$pendaturq; libra in |
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2295 eodem puncto C. & mo- |
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2296 ueatur libra in DE. mani |
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2297 <marg><I>Per def.c&etilde; tri grauitatis.</I></marg> fe$tum e$t libram non $o- |
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2298 lum in DE, $ed in quouis |
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2299 alio $itu manere. |
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2300 <fig> |
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2301 <pb n=29> |
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2302 <p>Sit autem centrum libræ |
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2303 AB $upra C in F; $itq; |
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2304 FC ip$i AB, & horizonti |
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2305 perpendicularis: & $i mo- |
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2306 ueatur libra in DE, linea |
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2307 CF mota erit in FG; quæ |
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2308 cùm non $it horizonti per- |
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2309 pendicularis, libra DE <marg>1 <I>Huius.</I></marg> |
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2310 deor$um ex parte D moue |
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2311 bitur, donec FG in FC |
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2312 redeat: atq; tunc libra DE |
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2313 in AB erit, in quò $itu |
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2314 quoq; manebit. |
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2315 <fig> |
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2316 <p>Et $i centrum libræ F |
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2317 $it infra libram; $itq; mota |
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2318 libra in DE; primùm qui |
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2319 dem manife$tum e$t li- |
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2320 bram in AB manere; in <marg>1. <I>Huius.</I></marg> |
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2321 DE verò deor$um ex par |
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2322 te E moueri: cùm linea |
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2323 FG non $it horizonti per- |
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2324 pendicularis. |
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2325 <fig> |
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2326 <p>Ex his determinatis $i libra $it |
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2327 arcuata, vel libræ brachia angulum |
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2328 con$tituant; centrumq; diuer$imo |
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2329 dè collocetur (quamquam hæc pro |
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2330 priè non $it libra) varios tamen |
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2331 huius quoq; effectus o$tendere pote |
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2332 rimus. Vt $it libra ACB, cuius |
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2333 centrum, circa quod vertitur, $it C. |
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2334 ductaq; AB, $it arcus $iue angulus |
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2335 <fig> |
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2336 ACB $upra lineam AB; & in AB grauitatis centra ponderum |
|
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2337 ponantur, quæ in hoc $itu maneant. moueatur deinde libra ab |
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2338 <foot>H</foot> |
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2339 <pb> |
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2340 hoc $itu, putá in ECF. Dico li- |
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2341 bram ECF in ACB redire. to- |
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2342 tius magnitudinis centrum grauita |
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2343 tis inueniatur D. & CD iunga- |
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2344 tur. Quoniam enim pondera AB |
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2345 <marg>1 <I>Huius.</I></marg> manent, linea CD horizonti per- |
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2346 pendicularis erit. quando igitur |
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2347 libra erit in ECF, linea CD erit |
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2348 putá in CG; quæ cùm non $it ho |
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2349 <fig> |
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2350 rizonti perpendicularis; libra ECF in ACB redibit. quod idem |
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2351 eueniet, $i centrum C $upra libram con$tituatur, vt in H. |
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2352 <p>Si verò arcus, $iue angulus |
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2353 ACB, $it infra lineam AB; eo |
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2354 dem modo libram ECF, cuius |
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2355 centrum, $iue $it in C, $iue in H, |
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2356 deor$um ex parte F moueri o- |
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2357 $tendemus. |
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2358 <fig> |
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2359 <p>Sit autem angulus ACB $upra lineam AB; aclibræ centrum |
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2360 $it H; lineaq; CH libram $u$tineat; & moueatur libra in EKF: |
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2361 libra EkF in ACB redibit. |
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2362 <pb n=30> |
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2363 <p>Si verò centrum libræ $it D, quocunq; modo moueatur libra; |
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2364 vbirelinquetur, manebit. |
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2365 <p>Si deinde punctum H $it infra lineam AB; tunc libra EkF |
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2366 deor$um ex parte F mouebitur. |
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2367 <p>Similiq; pror$us ratione, $i an |
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2368 gulus ACB $it infra lineam AB; |
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2369 $itq; libræ centrum H; $u$tineaturq; |
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2370 libralinea CH; $i libra ab hoc mo |
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2371 ueatur $itu, deor$um ex parte pon- |
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2372 deris inferioris mouebitur. & $i cen |
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2373 trum libræ $it D; vbi relinquetur, |
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2374 manebit. $i verò $it in K; $i ab eiu$ |
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2375 <fig> |
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2376 modi moueatur $itu, in eundem pro$us redibit. quæ omnia ex iis, |
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2377 quæ in principio diximus, $unt manife$ta. $imiliter $i centrum li |
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2378 bræ, vel in altero brachiorum, vel intra, vel extra vtcunq; po |
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2379 natur; eadem inueniemus. |
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2380 <foot>H 2</foot> |
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2381 <pb> |
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2382 <head>PROPOSITIO. V.</head> |
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2383 <p>Duo pondera in libra appen$a, $i libra inter |
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2384 hæc ita diuidatur, vt partes ponderibus per- |
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2385 mutatim re$pondeant; tàm in punctis appen$is |
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2386 ponderabunt, quàm $i vtraq; ex diui$ionis pun- |
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2387 cto $u$pendantur. |
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2388 <fig> |
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2389 <p>Sit AB libra, cuius centrum C; $intq; duo pondera EF ex pun |
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2390 ctis BG $u$pen$a: diuidaturq; BG in H, ita vt BH ad HG |
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2391 eandem habeat proportionem, quam pondus E ad pondus F. |
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2392 Dico pondera EF tàm in BG ponderare, quàm $i vtraq; ex pun |
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2393 cto H $u$pendantur. fiat AC ip$i CH æqualis. & vt AC ad |
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2394 CG, ita fiat pondus E ad pondus L. $imiliter vt AC ad CB, |
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2395 ita fiat pondus F ad pondus M. ponderaq; LM ex puncto A $u |
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2396 $pendantur. Quoniam enim AC e$t æqualis CH, erit BC ad |
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2397 CH vt pondus M ad pondus F. & quoniam maior e$t BC, |
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2398 quàm CH; erit & pondus M ip$o F maius. diuidatur igitur pon |
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2399 dus M in duas partes QR, $itq; pars Q ip$i F æqualis; erit BC |
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2400 <marg>17 <I>Quinti.</I></marg> ad CH, vt RQ ad Q: & diuidendo, vt BH ad HC, ita R ad Q. |
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2401 <marg><I>Cor.</I>4 <I>quinti.</I></marg> deinde conuertendo, vt CH ad HB, ita Q ad R. Præterea quo- |
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2402 niam CH e$t æqualis ip$i CA, erit HC ad CG, vt pondus |
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2403 E ad pondus L: maior autem e$t HC, quàm CG; erit & pon- |
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2404 <pb n=31> |
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2405 dus E pondere L maius. diuidatur itaq; pondus E in duas partes |
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2406 NO ita, vt pars O $it ip$i L æqualis, erit HC ad CG, vt to- |
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2407 tum NO ad O; & diuidendo, vt HG ad GC, ita N ad O: <marg>17 <I>Quinti. Cor.</I>4 <I>quin ti.</I></marg> |
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2408 conuertendoq; vt CG ad GH, ita O ad N. & iterum com- |
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2409 ponendo, vt CH ad HG, ita ON ad N. vt autem GH <marg>18 <I>Quinti.</I></marg> |
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2410 ad HB, ita e$t F ad ON. quare ex æquali, vt CH ad HB, ita F <marg>23 <I>Quinti.</I></marg> |
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2411 ad N. $ed vt CH ad HB ita e$t Q ad R: erit igitur Q ad R, vt <marg>11 <I>Quinti.</I></marg> |
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2412 F ad N; & permutando, vt Q ad F, ita R ad N. e$t autem pars <marg>16 <I>Quinti.</I></marg> |
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2413 Q ip$i F æqualis; quare & pars R ip$i N æqualis erit. Itaq; cùm |
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2414 pondus L $it ip$i O æquale, & pondus F ip$i Q etiam æquale, atq; |
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2415 pars R ip$i N æqualis; erunt pondera LM ip$is EF ponderibus |
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2416 æqualia. & quoniam e$t, vt AC ad CG, ita pondus E ad pon- |
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2417 dus L; pondera EL æqueponderabunt. $imiliter quoniam e$t, vt <marg>6 <I>Primi Ar chim. de æquep.</I></marg> |
|
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2418 AC ad CB, ita pundus F ad pondus M; pondera quoq; FM |
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2419 æqueponderabunt. Pondera igitur LM ponderibus EF in BG <marg>2 <I>Com. not. huius.</I></marg> |
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2420 appen$is æqueponderabunt. cùm autem di$tantia CA æqualis $it |
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2421 di$tantiæ CH; $i igitur vtraq; pondera EF in H appendantur, |
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2422 pondera LM ip$is EF ponderibus in H appen$is æquepondera- |
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2423 bunt. $ed LM ip$is EF in GB quoq; æqueponderant: æquè <marg>3 <I>Com not. huius.</I></marg> |
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2424 igitur grauia erunt pondera EF in GB, vt in H appen$a. tàm igi |
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2425 tur ponderabunt in BG, quàm in H appen$a. |
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2426 <fig> |
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2427 <p>Sint autem pondera EF in CB appen$a; $itq; C libræ centrum; |
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2428 & diuidatur CB in H, ita vt CH ad HB $it, vt pondus in F ad |
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2429 E. Dico pondera EF tàm in CB ponderare, quàm in puncto H. |
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2430 fiat CA ip$i CH æqualis, & vt CA ad CB, ita fiat pondus F ad |
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2431 aliud D, quod appendatur in A. Quoniam enim CH e$t æqua- |
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2432 <pb> |
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2433 <fig> |
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2434 lis CA, erit CH ad CB, vt F ad D; & maior quidem e$t CB, |
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2435 quàm CH; idcirco D pondere F maius erit. Diuidatur ergo D |
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2436 in duas partes Gk, $itq; G ip$i F æqualis; erit vt BC ad CH, |
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2437 vt Gk ad G; & diuidendo, vt BH ad HC, ita K ad G; & conuer |
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2438 <marg>17 <I>Quinti. Cor.</I>4 <I>quin ti.</I></marg> tendo, vt CH ad HB, ita G ad k. Vt autem CH ad HB, ita e$t |
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2439 <marg>11 <I>Quinti.</I></marg> F ad E. vt igitur G ad k, ita e$t F ad E; & permutando vt G |
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2440 <marg>16 <I>Quinti.</I></marg> ad F, ita k ad E. $unt autem GF æqualia; erunt & kE inter$e |
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2441 $e æqualia. cùm itaq; pars G $it ip$i F æqualis, & K ip$i E; erit |
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2442 totum C k ip$is EF ponderibus æquale. & quoniam AC e$t ip- |
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2443 $i CH æqualis; $i igitur pondera EF ex puncto H $u$pendantur, |
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2444 pondus D ip$is EF in H appen$is æqueponderabit. $ed & ip$is |
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2445 æqueponderat in CB, hoc e$t F in B, & E in C; cùm $it vt AC |
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2446 ad CB, ita F ad. D. pondus enim E ex centro libræ C $u$pen- |
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2447 $um non efficit, vt libra in alterutram moueatur partem. tàm igi- |
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2448 tur grauia erunt pondera EF in CB, quàm in H appen$a. |
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2449 <pb n=32> |
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2450 <fig> |
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2451 <p>Sit deniq; libra AB, & ex punctis AB $u$pen$a $int pondera |
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2452 EF; $itq; centrum libræ C intra pondera; diuidaturq; AB in |
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2453 D, ita vt AD ad DB $it, vt pondus F ad pondus E. Dico pon |
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2454 dera EF tàm in AB ponderare, quám $i vtraq; ex puncto D $u$pen |
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2455 dantur. fiat CG æqualis ip$i CD; & vt DC ad CA, ita fiat |
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2456 pondus E ad aliud H; quod appendatur in D. vt autem GC ad |
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2457 CB, ita fiat pondus Fad aliud K; appendaturq; k in G. Quoniã enim |
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2458 e$t, vt BC ad CG, hoc e$t ad CD, ita pondus k ad F; erit K ma |
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2459 ius pondere F. quare diuidatur pondus k in L, & MN; fiatq; |
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2460 pars L ip$i F æqualis; erit vt BC ad CD, vt totum LMN ad |
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2461 L; & diuidendo, vt BD ad DC, ita pars MN ad partem L. vt <marg>17 <I>Quinti.</I></marg> |
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2462 igitur BD ad DC, ita pars MN ad F. vt autem AD ad DB, |
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2463 ita F ad E: quare ex æquali, vt AD ad DC, ita MN ad E. cùm <marg>23 <I>Quinti.</I></marg> |
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2464 verò AD $it ip$a CD maior; erit & pars MN pondere E |
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2465 maior: diuidatur ergo MN in duas partes MN, $itq; M æqua |
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2466 lis ip$i E. erit vt AD ad DC, vt NM ad M; & diuidendo, vt <marg>17 <I>Quinti. Cor.</I>4 <I>quin ti</I></marg> |
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2467 AC ad CD, ita N ad M: conuertendoq; vt DC ad CA, ita M |
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2468 ad N. vt autem DC ad CA, ita e$t E ad H; erit igitur M ad N <marg>11 <I>Quinti.</I></marg> |
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2469 vt E ad H; & permutando, vt M ad E, ita N ad H. $ed ME <marg>16 <I>Quinti.</I></marg> |
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2470 $unt inter $e æqualia, erunt NH inter $e$e quoq; æqualia. & quo- |
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2471 niam ita e$t AC ad CD, vt H ad E: pondera HE æqueponde- <marg>6 <I>Primi Ar chim. de æquep.</I></marg> |
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2472 rabunt. $imiliter quoniam e$t vt GC ad CB, ita F ad k, ponde- |
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2473 <pb> |
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2474 <fig> |
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2475 <marg>2 <I>Com.not. huius.</I></marg> ra etiam kF æqueponderabunt. pondera igitur Ek HF in li- |
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2476 bra AB, cuius centrum C, æqueponderabunt. cùm autem GC |
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2477 ip$i CD $it æqualis, & pondus H $it ip$i N æquale; pondera NH |
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2478 æqueponderabunt. & quoniam omnia æqueponderant, demptis |
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2479 <marg>1 <I>Com.not. huius.</I></marg> HN ponderibus, quæ æqueponderant, reliqua æqueponderabunt; |
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2480 hoc e$t pondera EF & pondus LM ex centro libræ C $u$pen$a. |
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2481 quia verò pars L ip$i F e$t æqualis, & pars M ip$i E æqualis; erit |
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2482 totum LM ip$is FE ponderibus $imul $umptis æquale. & cùm |
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2483 $it CG ip$i CD æqualis, $i igitur pondera EF ex puncto D $u$pen- |
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2484 dantur, pondera EF in D appen$a ip$i LM æqueponderabunt. quare |
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2485 LM tàm ip$is EF in AB appen$is æqueponderat, quàm in pun |
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2486 cto D appen$is. libra enim $emper eodem modo manet. Ponde- |
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2487 <marg>3 <I>Com.not. huius.</I></marg> ra ergo EF tàm in AB ponderabunt, quàm in puncto D. quod |
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2488 demon$tre oportebat. |
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2489 <p>Hæc autem omnia (mechanicè tamen ma- |
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2490 gis) aliter o$tendemus. |
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2491 <pb n=33> |
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2492 <fig> |
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2493 <p>Sit libra AB, cuius centrum C; $intq; vt in primo ca$u duo pon |
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2494 dera EF ex punctis BG $u$pen$a: $itq; GH ad HB, vt pondus |
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2495 F ad pondus E. Dico pondera EF tàm in GB ponderare, quàm |
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2496 $i vtraq; ex diui$ionis puncto H $u$pendantur. Con$truantur ea |
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2497 dem, hoc e$t fiat AC ip$i CH æqualis, & ex puncto A duo ap- |
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2498 pendantur pondera LM, ita vt pondus E ad pondus L, $it vt |
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2499 CA ad CG; vt autem CB ad CA, ita $it pondus M ad pondus |
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2500 F. pondera LM ip$is EF in GB appen$is (vt $upra dictum e$t) |
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2501 æqueponderabunt. Sint deinde puncta NO centra grauitatis pon |
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2502 derum EF; conneci anturq; GN BO; iungaturq; NO, quæ tan- |
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2503 quam libra erit; quæ etiam efficiat lineas GN BO inter $e $e æqui- |
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2504 di$tantes e$$e; à punctoq; H horizonti perpendicularis ducatur |
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2505 HP, quæ NO $ecet in P, atq; ip$is GN BO $it æquidi$tans. |
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2506 deniq; connectatur GO, quæ HP $ecet in R. Quoniam igitur |
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2507 HR e$t lateri BO trianguli GBO æquidi$tans; erit GH ad HB, |
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2508 vt GR ad RO. $imiliter quoniam RP e$t lateri GN trianguli <marg>2 <I>Sexti.</I></marg> |
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2509 OGN æquidi$tans; erit GR ad RO, vt NP ad PO. quare |
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2510 vt GH ad HB, ita e$t NP ad PO. vt autem GH ad HB, ita <marg>11 <I>Quinti.</I></marg> |
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2511 e$t pondus F ad pondus E; vt igitur NP ad PO, ita e$t pondus |
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2512 F ad pondus E. punctum ergo P centrum erit grauitatis magni- |
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2513 tudinis ex vtri$q; EF ponderibus compo$itæ. Intelligantur itaq; <marg>6 <I>Primi Ar chim. de æquep.</I></marg> |
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2514 pondera EF ita e$$e à libra NO connexa, ac $i vna tantùm e$$et |
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2515 magnitudo ex vtri$q; EF compo$ita, in puncti$q; BG appen$a. $i |
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2516 igitur ponderum $u$pen$iones BG $oluantur, manebunt pondera <marg>1 <I>Huius.</I></marg> |
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2517 EF ex HP $u$pen$a; $icuti in GB prius manebant. pondera verò EF |
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2518 in GB appen$a ip$is LM ponderibus æqueponderant, & pondera |
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2519 <foot>I</foot> |
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2520 <pb> |
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2521 <fig> |
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2522 EF ex puncto H $u$pen$a, eandem habent con$titutionem ad li- |
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2523 bram AB, quam in BG appen$a: eadem ergo pondera EF ex |
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2524 H $u$pen$a ei$dem ponderibus LM æqueponderabunt. æquè igi- |
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2525 tur $unt grauia pondera EF in GB, vt in H appen$a. |
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2526 <fig> |
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2527 <p>Similiter demon$trabitur, pondera EF in quibu$cunq; aliis pun |
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2528 ctis appen$a tàm põderare, quàm $i vt raq; ex diui$ionis puncto H $u |
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2529 $pendantur. $i enim (vt $upra docuimus) in libra pondera inue- |
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2530 niantur, quibus pondera EF æqueponderent; eadem pondera EF |
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2531 ex H $u$pen$a ei$dem inuentis ponderibus æqueponderabunt; cùm |
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2532 punctum P $it $emper eorum centrum grauitatis; & HP horizon |
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2533 ri perpendicularis. |
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2534 <pb n=34> |
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2535 <head>PROPOSITIO. VI.</head> |
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2536 <p>Pondera æqualia in libra appen$a eam in gra |
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2537 uitate proportionem habent; quam di$tantiæ, ex |
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2538 quibus appenduntur. |
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2539 <fig> |
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2540 <p>Sit libra BAC $u$pen$a ex puncto A; & $ecetur AC vtcunq; |
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2541 in D: ex punctis autem DC appendantur æqualia pondera EF. |
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2542 Dico pondus F ad pondus E eam in grauitate proportionem ha- |
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2543 bere, quam habet di$tantia CA ad di$tantiam AD. fiat enim vt |
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2544 CA ad AD, ita pondus F ad aliud pondus, quod $it G. Dico pri |
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2545 múm pondera GF ex puncto C $u$pen$a tantùm ponderare, quan |
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2546 tùm pondera EF ex punctis DC. Secetur DC hifariam in H, & |
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2547 ex H appendantur vtraq; pondera EF. ponderabunt EF $imul |
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2548 $umpta in eo $itu, quantùm ponderant in DC. ponatur BA <marg>5 <I>Huius.</I></marg> |
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2549 æqualis AH, $eceturq; BA in K, ita vt $it KA æqualis AD: |
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2550 deinde ex puncto B appendatur pondus L duplum ponderis F, |
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2551 hoc e$t æquale duobus ponderibus EF, quod quidem æqueponde |
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2552 rabit ponderibus EF in H appen$is, hoc e$t appen$is in DC. Quoniã |
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2553 igitur, vt CA ad AD, ita e$t pondus F ad pondus G; erit compo |
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2554 nendo vt CA AD ad AD, hoc e$t vt Ck ad AD, ita ponde- <marg>18 <I>Quinti.</I></marg> |
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2555 ra FG ad pondus G. $ed cùm $it, vt CA ad AD, ita F pon- |
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2556 dus ad pondus G; erit conuertendo, vt DA ad AC, ita pondus <marg><I>Cor.</I> 4 <I>quinti.</I></marg> |
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2557 G ad pondus F; & con$equentium dupla, vt DA ad duplam ip$ius |
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2558 AC, ita pondus G ad duplum ponderis F, hoc e$t ad pondus |
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2559 L. Quare vt Ck ad DA, ita pondera EF ad pondus G; & vt |
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2560 <foot>I 2</foot> |
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2561 <pb> |
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2562 <fig> |
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2563 <marg>22 <I>Quinti.</I></marg> AD ad duplã ip$ius AC, ita pondus G ad pondus L; ergo ex æquali, |
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2564 vt Ck ad duplã ip$ius AC, ita pondera FG ad pondus L. $ed vt Ck |
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2565 ad duplam AC, ita dimidia CK, videlicet AH, hoc e$t BA, ad |
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2566 AC. Vt igitur BA ad AC, ita FG pondera ad pondus L. Qua |
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2567 re ex $exta eiu$dem primi Archimedis, duo pondera FG ex pun |
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2568 cto C $u$pen$a tantùm ponderabunt, quantùm pondus L ex B; |
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2569 hoc e$t quantùm pondera EF ex punctis DC $u$pen$a. Itaq; quo |
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2570 niam pondera FG tantùm ponderant, quantum pondera EF; $u- |
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2571 blato communi pondere F, tàm ponderabit pondus G in C ap- |
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2572 pen$um, quàm pondus E in D. ac propterea pondus F ad pon- |
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2573 <marg>7 <I>Quinti.</I></marg> dus E eam in grauitate proportionem habet, quam habet ad pon |
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2574 dus G. $ed pondus F ad G erat, vt CA ad AD:. ergo & F pon- |
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2575 dus ad pondus E eam in grauitate proportionem habebit, quam ha |
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2576 bet CA ad AD. quod demon$trare oportebat. |
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2577 <p>Si verò in libra |
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2578 BAC pondera EF |
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2579 æqualia ex punctis |
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2580 BC $u$pendantur; $i- |
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2581 militer dico pondus |
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2582 E ad pondus F eam |
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2583 <fig> |
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2584 in grauitate proportionem habere, quàm habet di$tantia CA ad di |
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2585 $tantiam AB. fiat AD ip$i AB æqualis, & ex puncto D $u$pen- |
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2586 datur pondus G æquale ponderi F; quod etiam ip$i E erit æquale. |
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2587 & quoniam AD e$t æqualis ip$i AB; pondera FG æqueponde |
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2588 rabunt, eandemq; habebunt grauitatem. cùm autem grauitas pon |
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2589 deris E ad grauitatem ponderis G $it, vt CA ad AD; erit graui |
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2590 tas ponderis E ad grauitatem ponderis F, vt CA ad AD, hoc e$t |
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2591 CA ad AB. quod erat quoq; o$tendendum. |
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2592 <pb n=35> |
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2593 <head>ALITER.</head> |
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2594 <p>Sit libra BAC, cu- |
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2595 ius centrum A; in pun- |
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2596 ctis verò BC pondera |
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2597 appendantur æqualia G |
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2598 F: $itq; primùm cen- |
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2599 trum A vtcunque inter |
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2600 BC. Dico pondus F ad |
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2601 pondus G eam in graui |
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2602 <fig> |
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2603 tate proportionem habere, quam habet di$tantia CA ad di$tan- |
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2604 tiam AB. fiat vt BA ad AC, ita pondus F ad aliud H, quod ap |
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2605 pendatur in B: pondera HF ex A æqueponderabunt. $ed cùm <marg>6 <I>Primi Ar chim. de æquep.</I></marg> |
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2606 pondera FG $int æqualia, habebit pondus H ad pondus G ean- |
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2607 dem proportionem, quam habet ad F. vt igitur CA ad AB, ita <marg>7 <I>Quinti.</I></marg> |
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2608 e$t H ad G. vt autem H ad G, ita e$t grauitas ip$ius H ad graui |
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2609 tatem ip$ius G; cùm in eodem puncto B $int appen$a. quare vt CA |
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2610 ad AB, ita grauitas ponderis H ad grauitatem ponderis G. cùm au |
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2611 tem grauitas ponderis F in C appen$i $it æqualis grauitati ponderis |
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2612 H in B; erit grauitas ponderis F ad grauitatem ponderis G, vt CA |
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2613 ad AB, videlicet vt di$tantia ad di$tantiam. quod demon$trare |
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2614 oportebat. |
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2615 <p>Si verò libra B |
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2616 AC $ecetur vtcunq; |
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2617 in D, & in DC ap- |
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2618 pendantur pondera |
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2619 æqualia EF. Dico |
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2620 $imiliter ita e$$e gra- |
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2621 <fig> |
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2622 uitatem ponderis F ad grauitatem ponderis E, vt di$tantia CA ad |
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2623 di$tantiam AD. fiat AB æqualis ip$i AD, & in B appendatur |
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2624 pondus G æquale ponderi E, & ponderi F. Quoniam enim AB e$t |
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2625 æqualis AD; pondera GE æqueponderabunt. $ed cùm grauitas |
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2626 ponderis F ad grauitatem ponderis G $it, vt CA ad AB, & graui |
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2627 tas ponderis E $it æqualis grauitati ponderis G; erit grauitas pon<*> |
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2628 deris F ad grauitatem ponderis E, vt CA ad AB, hoc e$t vt CA |
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2629 ad AD. qu od demon$trare oportebat. |
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2630 <pb> |
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2631 <head>COROLLARIVM.</head> |
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2632 <p>Ex hoc manife$tum e$t, quò pondus à centro |
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2633 libræ magis di$tat, eò grauius e$$e; & per con$e- |
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2634 quens velocius moueri. |
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2635 <p><marg><I>Stateræ ratio.</I></marg> Hinc præterea $tateræ quoq; ratio facilè o$ten |
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2636 detur. |
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2637 <p>Sit enim $tate |
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2638 ræ $capus AB, cu |
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2639 ius trutina $it in |
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2640 C; $itq; $tateræ |
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|
2641 appendiculum E. |
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2642 appendatur in A |
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2643 pondus D, quod |
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2644 æqueponderet ap |
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2645 pendiculo E in F |
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2646 <fig> |
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2647 appen$o. aliud quoq; appendatur pondus G in A, quod etiam |
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2648 appendiculo E in B appen$o æqueponderct. Dico grauitatem |
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2649 ponderis D ad grauitatem ponderis G ita e$$e, vt CF ad CB. |
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2650 Quoniam enim grauitas ponderis D e$t æqualis grauitati ponde- |
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2651 ris E in F appen$i, & grauitas ponderis G e$t æqualis grauitati pon |
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2652 deris E in B; erit grauitas ponderis D ad grauitatem ponderis E in |
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2653 F, vt grauitas ponderis G ad grauitatem ponderis E in B: & permu |
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2654 <marg>16 <I>Quinti.</I></marg> tando, vt grauitas ponderis D ad grauitatem ponderis G, ita graui |
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2655 tas ip$ius E in F, ad grauitatem ip$ius E in B; grauitas autem pon |
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2656 <marg>6 <I>Huius.</I></marg> deris E in F ad grauitatem ponderis E in B e$t, vt CF ad CB; vt |
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2657 igitur grauitas ponderis D ad grauitatem ponderis G, ita e$t CF |
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2658 ad CB $i ergo pars $capi CB in partes diuidatur æquales, $olo |
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2659 pondere E, & propius, & longius à puncto C po$ito; ponderum |
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2660 grauitates, quæ ex puncto A $u$penduntur inter $e $e notæ erunt. |
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2661 <pb n=36> |
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2662 Vt $i di$tantia CB tripla $it di$tantiæ CF, erit quoq; grauitas ip- |
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2663 $ius G grauitatis ip$ius D tripla, quod demon$trare oportebat. |
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2664 <p>Alio quoq; modo $tatera vti po$$umus, vt |
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2665 ponderum grauitates notæ reddantur. |
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2666 <p>Sit $capus AB, cuius tru- |
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2667 tina $it in C; $itq; $tateræ ap |
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2668 pendiculum E, quod appen- |
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2669 datur in A; $intqué pon- |
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2670 dera DG inæqualia, quorum |
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2671 inter $e $e grauitatum propor- |
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2672 tiones quærimus: appenda- |
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2673 tur pondus D in B, ita vt ip$i |
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2674 <fig> |
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2675 E æqueponderet. $imiliter pondus G appendatur in F, quod ei- |
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2676 dem ponderi E æqueponderet. dico D ad G ita e$$e, vt CF ad |
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2677 CB. Quoniam enim pondera DE æqueponderant, erit D ad E, <marg>6 <I>Primi Ar chim. de æquep.</I></marg> |
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2678 vt CA ad CB. cùm autem pondera quoque GE æquepon- |
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2679 derent, erit pondus E ad pondus G, vt FC ad CA; quare ex æqua |
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2680 li pondus D ad pondus G ita erit, vt CF ad CB. quod o$tende <marg>23 <I>Quinti.</I></marg> |
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2681 re quoq; oportebat. |
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2682 <pb> |
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2683 <head>PROPOSITIO VII.</head> |
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2684 <head>PROBLEMA.</head> |
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2685 <p>Quotcunque datis in libra ponderibus |
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2686 vbicunque appen$is, centrum libræ inuenire, |
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2687 ex quo $i $u$pendatur libra, data pondera ma- |
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2688 neant. |
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2689 <fig> |
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2690 <p>Sit libra AB, $intq; data quotcunque pondera CDEFG. |
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2691 accipiantur in libra vtcunque puncta AHkLB, ex quibus |
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2692 data pondera $pu$pendantur. Centrum libræ inuenire oportet, |
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2693 ex quo $i fiat $u$pen$io, data pondera maneant. Diuidatur |
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2694 <pb n=37> |
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2695 <fig> |
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2696 AH in M, ita vt HM ad MA, $it vt grauitas ponderis |
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2697 C ad grauitatem ponderis D. deinde diuidatur BL in N, ita |
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2698 vt LN ad NB, $it vt grauitas ponderis G ad grauitatem pon |
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2699 deris F. diuidaturq; MN in O, ita vt MO ad ON $it, vt |
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2700 granitas ponderum FG ad grauitatem ponderum CD. tandem- |
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2701 qué diuidatur kO in P, ita vt kP ad PO, $it vt grauitas pon |
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2702 derum CDFG ad grauitatem ponderis E. Quoniam igitur pon |
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2703 dera CDFG tàm ponderant in O, quàm CD in M, & FG in N; <marg>5 <I>Huius.</I></marg> |
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2704 æqueponderabunt pondera CD in M, & FG in N, & pondus E |
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2705 in K, $i ex puncto P $u$pendantur. cùm verò pondera CD tan |
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2706 tùm ponderent in M, quantùm in AH, & FG in N, quantùm |
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2707 in LB; pondera CDFG ex AHLB punctis $u$pen$a, & pon- |
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2708 dus E ex k, $i ex P $u$pendantur, æqueponderabunt, atq; mane- |
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2709 bunt. Inuentum e$t ergo centrum libræ P, ex quo data pondera |
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2710 manent. quod facere oportebat. |
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2711 <foot>K</foot> |
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2712 <pb> |
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2713 <head>COROLLARIVM.</head> |
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2714 <p>Ex hoc manife$tum e$t, $i ponderum CDEFG |
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2715 centra grauitatis e$$ent in AHKLB punctis; e$- |
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2716 $et punctum P magnitudinis ex omnibus CD |
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2717 EFG ponderibus compo$itæ centrum graui- |
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2718 tatis. |
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2719 <fig> |
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2720 <p>Hoc enim ex definitione centri grauitatis patet, cùm ponde- |
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2721 ra, $i ex puncto P $u$pendantur, maneant. |
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2722 <pb n=38> |
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2723 <head>DE VECTE.</head> |
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2724 <head>LEMMA.</head> |
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2725 <p>Sint quatuor magnitudines A |
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2726 BCD; $itq; A maior B, & C ma |
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2727 ior D. Dico A ad D maiorem |
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2728 habere proportionem; quàm |
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2729 habet B ad C. |
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2730 <p>Quoniam enim A ad C maiorem habet pro- |
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2731 portionem, quàm B ad C; & A ad D maio- <marg>8 <I>Quinti.</I></marg> |
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2732 rem quoq; habet proportionem, quam habet |
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2733 ad C: A igitur ad D maiorem habebit, quam B |
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2734 ad C. quod demon$trare oportebat. |
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2735 <fig> |
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2736 <head>PROPOSITIO I.</head> |
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2737 <p>Potentia $u$tinens pondus vecti appen$um; |
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2738 eandem ad ip$um pondus proportionem habe- |
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2739 bit, quam vectis di$tantia inter fulcimentum, ac |
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2740 ponderis $u$pen$ionem ad di$tantiam à fulcimen |
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2741 to ad potentiam interiectam. |
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2742 <foot>K 2</foot> |
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2743 <pb> |
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2744 <fig> |
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2745 <p>Sit vectis AB, cuius fulcimeutum C; $itq; pondus D ex A $u- |
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2746 $pen$um AH, ita vt AH $it $emper horizonti perpendicularis: |
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2747 $itq; potentia $u$tinens pondus in B. Dico potentiam in B ad pon |
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2748 dus D ita e$$e, vt CA ad CB. fiat vt BC ad CA, ita pondus D |
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2749 <marg>6 <I>Primi Ar chim. de æquep.</I></marg> ad aliud pondus E, quippè quod $i in B appendatur; ip$i D æque |
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2750 ponderabit, exi$tente C amborum grauitatis centro. quare poten |
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2751 tia æqualis ip$i E ibidem con$tituta ip$i D æqueponderabit, vecte |
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2752 AB, eius fulcimento in C collocato, hoc e$t prohibebit, ne pon |
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2753 dus D deor$um vergat, quemadmodum prohibet pondus E. Po |
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2754 <marg><I>Ex</I> 7 <I>quinti.</I></marg> tentia verò in B ad pondus D eandem habet proportionem, quam |
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2755 pondus E ad idem pondus D: ergo potentia in B ad pondus D |
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2756 erit, vt CA ad CB; hoc e$t vectis di$tantia à fulcimento ad pon |
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2757 deris $u$pendium ad di$tantiam à fulcimento ad potentiam. quod |
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2758 demon$trare oportebat. |
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2759 <p>Hinc facilè o$tendi pote$t, fulcimentum quò |
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2760 ponderi fuerit propius, minorem ad idem pon- |
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2761 dus $u$tinendum requiri potentiam. |
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2762 <p>Ii$dem po$i- |
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2763 tis, $it fulcimen |
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2764 tum in F ip$i A |
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2765 propius, quàm |
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2766 C; fiatq; vt BF |
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2767 ad FA, ita pon |
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2768 dus D ad aliud |
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2769 <fig> |
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2770 G, quod $i appendatur in B, pondera DG ex fulcimento E |
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2771 <marg><I>Ex eadem Sexta.</I></marg> æqueponderabunt. quoniam autem BF maior e$t BC, & CA |
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2772 <marg><I>Lemma.</I></marg> maior AC; maior erit proportio BF ad FA, quàm BC ad CA: |
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2773 <pb n=39> |
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2774 & ideo maior quoq; erit proportio ponderis D ad pondus G, |
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2775 quàm idem D ad E: pondus igitur G minus erit pondere E. cùm <marg>10 <I>Quinti.</I></marg> |
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2776 autem potentia in B ip$i G æqualis ponderi D æqueponderet, mi- |
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2777 nor potentia, quàm ea, quæ ponderi E e$t æqualis, pondus D $u |
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2778 $tinebit; exi$tente vecte AB, eius verò fulcimento vbi F, quàm $i |
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2779 fuerit vbi C. $imiliter quoq; o$tendetur, quò propius erit fulci- |
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2780 mentum ponderi D, adhuc $emper minorem requiri potentiam |
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2781 ad $u$tinendum pondus D. |
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2782 <head>COROLLARIVM.</head> |
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2783 <p>Vnde palàm colligere licet, exi$tente AF ip$a |
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2784 FB minore, minorem quoq; requiri potentiam |
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2785 in ip$o B pondere D $u$tinendo. æquali verò |
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2786 æqualem. maiore verò maiorem. |
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2787 <head>PROPOSITIO II.</head> |
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2788 <p>Alio modo vecte vti po$sumus. |
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2789 <p>Sit vectis AB, cuius |
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2790 fulcimentum $it B, & |
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2791 pondus C vtcunq; in |
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2792 D inter AB appen- |
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2793 $um; $itq; potentia in |
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2794 A $u$tinens pondus C. |
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2795 Dico vt BD ad BA, |
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2796 <fig> |
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2797 ita e$$e potentiam in A ad pondus C. appendatur in A pondus |
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2798 E æquale ip$i C; & vt AB ad BD, ita fiat pondus E ad aliud F. |
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2799 & quoniam pondera CE $unt inter $e $e æqualia, erit pondus C |
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2800 ad pondus F, vt AB ad BD. appendatur quoq; pondus F in A. |
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2801 & quoniam pondus E ad pondus F e$t, vt grauitas ip$ius E ad gra- <marg><I>In $exta hu ius de libra Ex</I> 11 <I>quin ti.</I></marg> |
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2802 uitatem ip$ius F; & pondus E ad F e$t, vt AB ad BD; vt igitur |
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2803 grauitas ponderis E ad grauitatem ponderis F, ita e$t AB ab BD. |
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2804 vt autem AB ad BD, ita e$t grauitas ponderis E ad grauitatem <marg>6 <I>Huius. de libra.</I></marg> |
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2805 <pb> |
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2806 ponderis C: quare gra |
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2807 uitas ponderis E ad |
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2808 grauitatem ponderis |
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2809 F ita erit, vt grauitas |
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2810 ponderis E ad gra- |
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2811 uitatem ponderis C. |
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2812 Pondera igitur CF |
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2813 <fig> |
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2814 <marg><I>Ex</I> 9 <I>quinti.</I></marg> eandem habent grauitatem. Ponatur itaq; potentia in A $u$tinens |
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2815 pondus F; erit potentia in A æqualis ip$i ponderi F. & quoniam |
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2816 pondus F in A appen$um æquè graue e$t, vt pondus C in D ap- |
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2817 pen$um; eandem proportionem habebit potentia in A ad grauita- |
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2818 <marg><I>Ex</I> 7 <I>quinti.</I></marg> tem ponderis F in A appen$i, quam habet ad grauitatem ponde- |
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2819 ris C in D appen$i. Potentia verò in A ip$i F æqualis $u$tinet |
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2820 pondus F, ergo potentia in A pondus quoq; C $u$tinebit. Itaq; |
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2821 cùm potentia in A $it æqualis ponderi F, & pondus C ad pon- |
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2822 dus F $it, vt AB ad BD; erit pondus C ad potentiam in A, vt |
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2823 <marg><I>Cor.</I> 4 <I>quin ti.</I></marg> AB ad BD. & è conuer$o, vt BD ad BA, ita potentia in A ad |
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2824 pondus C. potentia ergo ad pondus ita erit, vt di$tantia fulci- |
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2825 mento, ac ponderis $u$pen$ioni intercepta ad di$tantiam à fulci |
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2826 mento ad potentiam. quod oportebat demon$trare. |
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2827 <head>ALITER.</head> |
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2828 <fig> |
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2829 <p>Sit vectis AB, cuius fulcimentum $it B, & pondus E ex puncto |
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2830 C $u$pen$um; $itq; vis in A $u$tinens pondus E. Dico vt BC ad BA, |
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2831 ita e$$e potentiam in A ad pondus E. Producatur AB in C, & |
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2832 fiat BD æqualis BC; & ex puncto D appendatur pondus F æqua |
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2833 le ponderi E; itemq; ex puncto A $u$pendatur pondus G ita, vt |
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2834 pondus F ad pondus G eandem habeat proportionem, quam AB |
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2835 <pb n=40> |
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2836 ad BA. pondera FG æqueponderabunt. cùm autem $it CB æqua |
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2837 lis BD, pondera quoq; FE æqualia æqueponderabunt. pondera |
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2838 verò FEG in libra, $euvecte DBA appen$a, cuius fulcimentum |
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2839 e$t B, non æqueponderabunt; $ed ex parte A deor$um tendent. po |
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2840 natur itaq; in A tanta vis, vt pondera FEG æqueponderent; erit |
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2841 potentia in A æqualis ponderi G. pondera enim FE æqueponderãt, |
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2842 & visin A nihil aliud efficere debet, ni$i $u$tinere põdus G, ne de$cen |
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2843 dat. & quoniam pondera FEG, & potentia in A æqueponderant, |
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2844 demptis igitur FG ponderibus, quæ æqueponderant, reliqua æque |
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2845 ponderabunt; $cilicet potentia in A ponderi E, hoc e$t potentia |
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2846 in A pondus E $u$tinebit, ita vt vectis AB maneat, vt prius erat. |
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2847 Cùm autem potentia in A $it æqualis ponderi G, & pondus E pon |
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2848 deri F æquale; habebit potentia in A ad pondus E eandem pro- |
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2849 portionem, quam habet BD, hoc e$t BC ad BA. quod demon- |
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2850 $trare oportebat. |
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2851 <head>COROLLARIVM I.</head> |
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2852 <p>Ex hoc etiam (vt prius) manife$tum e$$e po- |
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2853 te$t, $i ponatur pondus E propius fulcimento B, |
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2854 vt in H; minorem potentiam in A $u$tinere po$- |
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2855 $e ip$um pondus. |
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2856 <p>Minorem enim proportionem habet HB ad BA, quam CB ad <marg>8 <I>Quinti.</I></marg> |
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2857 BA. & quò propius pondus erit fulcimento, adhuc $emper mino |
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2858 rem po$$e potentiam $u$tinere pondus E $imiliter o$tendetur. |
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2859 <head>COROLLARIVM II.</head> |
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2860 <p>Sequitur etiam potentiam in A $emper mino |
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2861 rem e$$e pondere E. |
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2862 <p>Sumatur enim inter AB quoduis punctum C, $emper BC |
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2863 minor erit BA. |
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2864 <pb> |
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2865 <head>COROLLARIVM III.</head> |
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2866 <p>Ex hoc quoq; elici pote$t, $i duæ fuerint poten |
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2867 tiæ, vna in A, altera in B, & vtraq; $u$tentet |
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2868 pondus E; potentiam in A ad potentiam in B e$- |
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2869 $e, vt BC ad CA. |
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2870 <p>Vectis enim BA fungi- |
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2871 tur officio duorum vectiũ; |
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2872 & AB $unt tanquam duo |
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2873 fulcimenta, hoc e$t quan- |
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2874 do AB e$t vectis, & poten |
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2875 tia $u$tinens in A; erit eius |
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2876 <fig> |
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2877 fulcimentum B. Quando verò BA e$t vectis, & potentia in B; |
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2878 erit A fulcimentum: & pondus $emper ex puncto C remanet $u- |
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2879 $pen$um. & quoniam potentia in A ad pondus E e$t, vt BC ad |
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2880 BA; vt autem pondus E ad potentiam, quæ e$t in B, ita e$t |
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2881 <marg>22 <I>Quinti.</I></marg> BA ad AC; erit ex æquali, potentia in A ad potentiam in B, vt |
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2882 BC ad CA. & hoc modo facilè etiam proportionem, quæ in |
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2883 Quæ$tionibus Mechanicis quæ$tione vige$ima nona ab Ari$totele |
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2884 ponitur, noui$$e poterimus. |
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2885 <head>COROLLARIVM IIII.</head> |
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2886 <p>E$t etiam manife$tum, vtra$q; potentias in A, |
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2887 & B $imul $umptas æquales e$$e ponderi E. |
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2888 <p>Pondus enim E ad potentiam in A e$t, vt BA ad BC; & idem |
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2889 pondus E ad potentiam in B e$t, vt BA ad AC; quare pondus |
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2890 E ad vtra$q; potentias in A, & B $imul $umptas e$t, vt AB ad BC |
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2891 CA $imul, hoc e$t ad BA. pondus igitur E vtri$q; potentiis $imul |
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2892 $umptis æquale erit. |
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|
2893 <pb n=41> |
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|
2894 <head>PROPOSITIO III.</head> |
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2895 <p>Alio quoq; modo vecte vti po$sumus. |
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|
2896 <p>Sit Vectis AB, |
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2897 cuius fulcimentum |
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|
2898 B; $itq; ex puncto |
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|
2899 A pondus C appen- |
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|
2900 $um; $itq; potentia |
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|
2901 in D vtcunq; inter |
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|
2902 AB $u$tinens pon- |
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2903 dus C. Dico vt AB |
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|
2904 <fig> |
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|
2905 ad BD, ita e$$e potentiam in D ad pondus C. Appendatur ex |
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2906 puncto D pondus E æquale ip$i C; & vt BD ad BA, ita fiat pon |
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2907 dus E ad aliud F. & cùm pondera CE $int inter $e $e æqualia; erit |
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2908 pondus C ad pondus F, vt BD ad BA. appendatur pondus |
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2909 F quoq; in D. & quoniam pondus E ad ip$um F e$t, vt grauitas |
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2910 ponderis E ad grauitatem ponderis F; & pondus E ad pondus F <marg><I>In $exta hu ius de libra.</I></marg> |
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2911 e$t, vt BD ad BA: vt igitur grauitas ponderis E ad grauitatem |
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2912 ponderis F, ita e$t BD ad BA. vt autem BD ad BA, ita e$t gra <marg>6 <I>Huius de libra.</I></marg> |
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2913 uitas ponderis E ad grauitatem ponderis C; quare grauitas ponde- |
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2914 ris E ad grauitatem ponderis F eandem habet proportionem, |
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2915 quam habet ad grauitatem ponderis C. pondera ergo CF eandem <marg>9 <I>Quinti.</I></marg> |
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2916 habent grauitatem. $it igitur potentia in D $u$tinens pondus F, |
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2917 erit potentia in D ip$i ponderi F æqualis. & quoniam pondus F |
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2918 in D æquè graue e$t, vt pondus C in A; habebit potentia in D |
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2919 eandem proportionem ad grauitatem ponderis F, quam habet ad <marg>7 <I>Quinti.</I></marg> |
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2920 grauitatem ponderis C. $ed potentia in D pondus F $u$tinet; po- |
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2921 tentia igitur in D pondus quoq; C $u$tinebit: & pondus C ad po- |
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2922 tentiam in D ita erit, vt pondus C ad pondus F; & C ad F e$t, vt |
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2923 BD ad BA; erit igitur pondus C ad potentiam in D, vt BD ad |
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2924 BA: & conuertendo, vt AB ad BD, ita potentia in D ad pondus |
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2925 C. potentia ergo ad pondus e$t, vt di$tantia à fulcimento ad pon |
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2926 deris $u$pendium ad di$tantiam à fulcimento ad potentiam. quod |
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2927 demon$trare oportebat. |
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2928 <foot>L</foot> |
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2929 <pb> |
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2930 <head>ALITER.</head> |
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2931 <fig> |
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2932 <p>Sit vectis AB, cuius fulcimentum B; & ex puncto A $it pon- |
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2933 dus C $u$pen$um; $itq; potentia in D $u$tinens pondus C. Dico |
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2934 vt AB ad BD, ita e$$e potentiam in D ad pondus C. Produca |
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2935 tur AB in E, fiatq; BE æqualis ip$i BA; & ex puncto E appen |
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2936 datur pondus F æquale ponderi C; & vt BD ad BE, ita fiat pon |
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2937 dus F ad aliud G, quod ex puncto D $u$pendatur. pondera FG |
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2938 æqueponderabunt. & quoniam AB e$t æqualis BE, & pondera |
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2939 FC æqualia; $imiliter pondera FC æqueponderabunt. Pondera |
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2940 verò FGC $u$pen$a in vecte EBA, cuius fulcimentum e$t B, non |
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2941 æqueponderabunt; $ed ex parte A deor$um tendent. Ponatur igi |
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2942 tur in D tanta vis, vt pondera FGC æqueponderent; erit po- |
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2943 tentia in D æqualis ponderi G: pondera enim FC æquepon de- |
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2944 rant, & potentia in D nil aliud efficere debet, ni$i $u$tinere pon- |
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2945 dus G ne de$cendat. & quoniam pondera FGC, & potentia in |
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2946 D æqueponderant, demptis igitur FG ponderibus, quæ æquepon |
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2947 derant; reliqua æqueponderabunt, $cilicet potentia in D ponderi C. |
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2948 hoc e$t potentia in D pondus C $u$tinebit, ita vt vectis AB ma- |
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2949 neat, vt prius. & cùm potentia in D $it æqualis ponderi G, & pon- |
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2950 dus C æquale ponderi F; habebit potentia in D ad pondus C ean |
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2951 dem proportionem, quam EB, hoc e$t AB ad BD. quod de- |
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2952 mon$trare oportebat. |
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|
2953 <head>COROLLARIVM I.</head> |
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2954 <p>Ex hoc etiam pàtet, vt prius, $i coftituatur pon |
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2955 dus fulcimento B propius, vt in H; à minori po- |
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2956 tentia pondus ip$um $ub$tineri debere. |
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|
2957 <pb n=42> |
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2958 <p>Minorem enim proportionem habet HB ad BD, quàm AB ad <marg>8 <I>Quinti.</I></marg> |
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2959 BD. & quò propius erit fulcimento, adhuc $emper minorem re- |
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2960 quiri potentiam. |
|
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|
2961 <head>COROLLARIVM II.</head> |
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2962 <p>Manife$tum quoq; e$t, potentiam in D $emper |
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2963 maiorem e$$e pondere C. |
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2964 <p>Si enim inter AB $umatur quoduis punctum D, $emper AB |
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2965 maior erit BD. |
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2966 <p>Et aduertendum e$t ha$ce, quas attulimus demon$trationes |
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2967 non $olum vectibus horizonti æquidi$tantibus, verùm etiam ve- |
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2968 ctibus horizonti inclinatis ad hæc omnia o$tendenda commodè |
|
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2969 aptari po$$e. quod ex iis, quæ de libra diximus, patet. |
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|
2970 <head>PROPOSITIO IIII.</head> |
|
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2971 <p>Si potentia pondus in vecte appen$um mo- |
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2972 ueat; erit $patium potentiæ motæ ad $patium |
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2973 moti ponderis, vt di$tantia à fulcimento ad po- |
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2974 tentiam ad di$tantiam ab eodem ad ponderis $u |
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2975 $pen$ionem. |
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2976 <foot>L 2</foot> |
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2977 <pb> |
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2978 <p>Sit vectis AB, cuius ful- |
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2979 cimentum C; & ex puncto B |
|
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2980 $it pondus D $u$pen$um; $itq; |
|
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|
2981 potentia in A mouens pon- |
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2982 dus D vecte AB. Dico $pa- |
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2983 tium potentiæ in A ad $pa- |
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2984 tium ponderis ita e$$e, vt CA |
|
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2985 ad CB. Moueatur vectis AB, |
|
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|
2986 & vt pondus D $ur$um mo- |
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2987 ueatur, oportet B $ur$um mo |
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2988 ueri, A verò deor$um. & quo- |
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2989 niam C e$t punctum immobi |
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2990 le; idcirco dum A, & B mo- |
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2991 uentur, circulorũ circumferen |
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2992 tias de$cribent. Moueatur igi- |
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2993 tur AB in EF; erunt AE |
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2994 <fig> |
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2995 BF circulorum circumferentiæ, quorum $emidiametri $unt CA |
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2996 CB. tota compleatur circumferentia AGE, & tota BHF; $intq; |
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2997 KH puncta, vbi AB, & EF circulum BHF $ecant. Quoniam e- |
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2998 <marg>15 <I>Primi.</I></marg> nim angulus BCF e$t æqualis angulo HCk; erit circumferentia |
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2999 <marg><I>Ex</I> 26 <I>tertii.</I></marg> kH circumferentiæ BF æqualis. cùm autem circumferentiæ AE |
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3000 kH $int $ub eodem angulo ACE, & circumferentia AE ad to- |
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3001 tam circumferentiam AGE $it, vt angulus ACE ad quatuor re- |
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3002 ctos; vt autem idem angulus HCk ad quatuor rectos, ita quoq; |
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3003 e$t circumferentia HK ad totam circumferentiam HBK; erit cir |
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3004 cumferentia AE ad totam circumferentiam AGE, vt circumfe- |
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3005 <marg>16 <I>Quinti.</I></marg> rentia kH ad totam kFH. & permutando, vt circumferentia |
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3006 AE ad circumferentiam kH, hoc e$t BF, ita tota circumferen- |
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3007 tia AGE ad totam circumferentiam BHF. totaverò circumfe |
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3008 rentia AGE ita $e habet ad totam BHF, vt diameter circuli AEG |
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3009 <marg>23 <I>Octaui Pappi.</I></marg> ad diametrum circuli BHF. Vt igitur circumferentia AE ad cir |
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3010 <marg>11 <I>Quinti.</I></marg> cumferentiam BF, ita diameter circuli AGE ad diametrum cir |
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3011 culi BHF: vt autem diameter ad diametrum, ita $emidiameter |
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3012 ad $emidiametrum, hoc e$t CA ad CB: quare vt circumferen- |
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3013 tia AE ad circumferentiam BF, ita CA ad CF. circumferentia |
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3014 verò AE $patium e$t potentiæ motæ, & circumferentia BF e$t |
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|
3015 <pb n=43> |
|
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3016 æqualis $patio ponderis D moti. $patium enim motus ponderis |
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3017 D $emper æquale e$t $patio motus puncti B, cùm in B $it appen |
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3018 $um: $patium ergo potentiæ motæ ad $patium moti ponderis e$t, |
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3019 vt CA ad CB; hoc e$t vt di$tantia à fulcimento ad potentiam |
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3020 ad di$tantiam ab eodem ad ponderis $u$pen$ionem. quod demon |
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|
3021 $trare oportebat. |
|
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3022 <p>Sit autem vectis AB, cu- |
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3023 ius fulcimentum B; potentia- |
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|
3024 qué mouens in A; & pondus |
|
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|
3025 in C. dico $patium potentiæ |
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|
3026 translatæ ad $patium transla |
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|
3027 ti ponderis ita e$$e, vt BA ad |
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|
3028 BC. Moueatur vectis, & vt |
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3029 pondus $uríum attollatur, ne- |
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3030 ce$$e e$t puncta C A $ur$um |
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|
3031 moueri. Moueatur igitur A |
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3032 $ur$um v$q; ad D; $itq; ve- |
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|
3033 ctis motus BD. eodemq; |
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|
3034 modo (vt prius dictum e$t) |
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3035 o$tendemus puncta CA cir- |
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|
3036 culorum circumferentias de- |
|
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|
3037 <fig> |
|
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3038 $cribere, quorũ $emidiametri $unt BA BC. $imiliterq; o$tendemus |
|
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|
3039 ita e$$e AD ad CE, vt $emidiameter AB ad $emidiametrum BC. |
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|
3040 <p>Eademq; ratione, $i potentia e$$et in C, & pondus in A, |
|
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3041 o$tendetur ita e$$e CE ad AD, vt BC ad BA; hoc e$t di$tan |
|
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|
3042 tia à fulcimento ad potentiam ad di$tantiam ab eodem ad ponde |
|
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|
3043 ris $u$pen$ionem. quod oportebat demon$trare. |
|
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|
3044 <head>COROLLARIVM.</head> |
|
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3045 <p>Ex his manife$tum e$t maiorem habere pro- |
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|
3046 portionem $patium potentiæ mouentis ad $pa- |
|
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3047 tium ponderis moti, quàm pondus ad eandem |
|
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|
3048 potentiam. |
|
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3049 <p>Spatium enim potentiæ ad $patium ponderis eandem habet, |
|
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|
3050 <pb> |
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|
3051 quam pondus ad potentiam pondus $u$tinentem; potentia ve- |
|
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|
3052 rò $u$tinens minor e$t potentia mouente, quare minorem habebit |
|
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|
3053 <marg>8 <I>Quinti.</I></marg> proportionem pondus ad potentiam ip$um mouentem, quàm ad |
|
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|
3054 potentiam ip$um $u$tinentem. $patium igitur potentiæ mouentis |
|
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|
3055 ad $patium ponderis maiorem habebit proportionem, quàm pon- |
|
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|
3056 dus ad candem potentiam. |
|
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|
3057 <head>PROPOSITIO V.</head> |
|
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|
3058 <p>Potentia quomodocunq; vecte pondus $u$ti- |
|
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|
3059 nens ad ip$um pondus eandem habebit propor- |
|
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|
3060 tionem, quam di$tantia à fulcimento ad punctum, |
|
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|
3061 vbi à centro grauitatis ponderis horizonti ducta |
|
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|
3062 perpendicularis vectem $ecat, intercepta, ad |
|
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|
3063 di$tantiam inter fulcimentum, & potentiam. |
|
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|
3064 <p>Sit vectis AB |
|
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|
3065 horizonti æqui- |
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|
3066 di$tans, cuius ful |
|
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|
3067 cimentum N; $it |
|
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|
3068 deinde pondus |
|
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|
3069 AC, cuius cen- |
|
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|
3070 trum grauitatis |
|
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|
3071 $it D, quod pri |
|
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|
3072 mùm $it infra ve |
|
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|
3073 ctem; pondus ve |
|
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|
3074 rò $it ex punctis |
|
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|
3075 AO $u$pen$um; |
|
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|
3076 <fig> |
|
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|
3077 & à puncto D horizonti, & ip$i AB perpendicularis ducatur DE. |
|
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|
3078 $i verò alii $int quoq; vectes AF AG, quorum fulcimenta $int |
|
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|
3079 HK; pondu$q; AC in vecte AG ex punctis AQ $it appen$um; |
|
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|
3080 in vecte autem AF in punctis AP: lineaq; DE producta $ecet |
|
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|
3081 AF in L, & AG in M. dico potentiam in F pondus AC $u$tinen |
|
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|
3082 tem ad ip$um pondus eam habere proportionem, quam habet kL |
|
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|
3083 <pb n=44> |
|
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|
3084 ad kF; & potentiam in B ad pondus eam habere, quam NE ad |
|
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|
3085 NB; & potentiam in G ad pondus eam, quam HM ad HG. |
|
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|
3086 Quoniam enim DL horizonti e$t perpendicularis, pondus AC |
|
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|
3087 vbicunq; in linea DL fuerit appen$um, eodem modo, quo reperi- |
|
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|
3088 tur, manebit. quare in vecte AB $i $u$pen$iones, quæ $unt ad AO |
|
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|
3089 $oluantur, pondus AC in E appen$um eodem modo manebit, $i- |
|
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|
3090 cutinunc manet; hoc e$t $ublato puncto A, & linea QO, codem |
|
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|
3091 modo pondus in E appen$um manebit, vt ab ip$is AO pun- |
|
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|
3092 ctis $u$tinebatur; ex commentario Federici Commandini in $extam |
|
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|
3093 Archimedis propo$ion&etilde; de quadratura parabolæ, & ex prima huius |
|
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|
3094 de libra. Itaq; quoniam pondus AC eandem ad libram habet con$ti |
|
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|
3095 tutionem, $iue in AO $u$tineatur, $iue ex puncto E $it appen$um; |
|
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|
3096 eadem potentia in B idem pondus AC, $iue in E, $iue in AO |
|
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|
3097 $u$pen$um $u$tinebit. potentia verò in B $u$tinens pondus AC |
|
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|
3098 in E appen$um ad ip$um pondus ita $e habet, vt NE ad NB; po- <marg>1 <I>Huius.</I></marg> |
|
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|
3099 tentia igitur in B $u$tinens pondus AC ex punctis AO $u$pen |
|
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|
3100 $um ad ip$um pondus ita erit, vt NE ad NB. Non aliter o$ten |
|
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|
3101 detur pondus AC ex puncto L $u$pen$um manere, $icuti à pun |
|
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|
3102 ctis AP $u$tinetur; potentiamq; in F ad ip$um pondus ita e$$e, vt kL |
|
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|
3103 ad KF. In vecte verò AG pondus AC in M appen$um ita mane |
|
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|
3104 re, vt à punctis AQ $u$tinetur; potentiamq; in G ad pondus |
|
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|
3105 AC ita e$$e, vt HM ad HG; hoc e$t vt di$tantia à fulcimento |
|
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|
3106 ad punctum, vbi à centro grauitatis ponderis horizonti ducta |
|
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|
3107 perpendicularis vectem $ecat, ad di$tantiam à fulcimento ad poten |
|
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|
3108 tiam. quod demon$trare oportebat. |
|
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|
3109 <p>Si autem FBG e$$ent vectium fulcimenta, potentiæq; e$$ent |
|
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|
3110 in KNH pondus $u$tinentes, $imili modo o$tendetur ita e$$e po |
|
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|
3111 tentiam in H ad pondus, vt GM ad GH; & potentiam in N ad |
|
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|
3112 pondus, vt BE ad BN; ac potentiam in k ad pondus, vt FL |
|
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|
3113 ad Fk. |
|
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|
3114 <pb> |
|
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|
3115 <p>Et $i vectes AB |
|
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|
3116 AF AG habeant |
|
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|
3117 fulcimenta in A, |
|
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|
3118 & pondus $it NO; |
|
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|
3119 deinde ab eius |
|
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|
3120 centro grauitatis |
|
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|
3121 D ducatur ip$i A |
|
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|
3122 B, & horizonti |
|
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|
3123 perp&etilde;dicularis D |
|
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|
3124 MEL; $intq; po |
|
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|
3125 tentiæ in FBG: |
|
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|
3126 $imiliter o$tende- |
|
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|
3127 tur ita e$$e poten- |
|
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|
3128 <fig> |
|
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|
3129 tiam in G pondus NO $u$tinentem ad ip$um pondus, vt AM |
|
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|
3130 ad AG; ac potentiam in B, vt AE ad AB; & potentiam in F, |
|
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|
3131 vt AL ad AF. |
|
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|
3132 <p>Sit deinde |
|
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|
3133 vectis AB ho |
|
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|
3134 rizonti æqui- |
|
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|
3135 di$tans, cuius |
|
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|
3136 fulcimentum |
|
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|
3137 D; & $it BE |
|
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|
3138 pondus, cuius |
|
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|
3139 centrum gau i |
|
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|
3140 tatis $it F $u- |
|
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|
3141 pra vectem: à |
|
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|
3142 punctoq; F ho |
|
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|
3143 rizonti, & ip$i |
|
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|
3144 AB ducatur |
|
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|
3145 <fig> |
|
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|
3146 FH; pondu$q; à puncto B, & PQ $u$tineatur. Sint deinde alii ve- |
|
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|
3147 ctes BL BM, quorum fulcimenta $int NO; lineaq; FH producta $e- |
|
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|
3148 cet BM in k, & BL in G; pondus autem in vecte BL in pun- |
|
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|
3149 ctis BP $u$tineatur; in vecte autem BM à puncto B, & PR. Di- |
|
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|
3150 co potentiam in L pondus BE vecte BL $u$tinentem ad ip$um |
|
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|
3151 pondus eam habere proportionem, quam NG ad NL; & po- |
|
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|
3152 <pb n=45> |
|
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|
3153 tentiam in A ad pondus eam habere, quam DH ad DA; poten |
|
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|
3154 tiamq; in M ad pondus eam, quam Ok ad OM. Quoniam e- |
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3155 nim à centro grauitatis F ducta e$t kF horizonti perpendicularis, |
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3156 ex quocunq; puncto lineæ kF $u$tineatur pondus, manebit; vt <marg>1 <I>Huius de libra.</I></marg> |
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3157 nunc $e habet. $i igitur $u$tineatur in H, manebit vt prius; $cili- |
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3158 cet $ublato puncto B, & PQ, quæ pondus $u$tinent, pondus BE |
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3159 manebit, $icuti ab ip$is $u$tinebatur. quare in vecte AB graue$cet |
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3160 in H, & ad vectem eandem habebit con$titutionem, quam prius; |
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3161 idcirco erit, ac $i in H e$$et appen$um. eadem igitur potentiaìdem |
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3162 pondus BE, $iue in H, $iue in B, & Q $uffultum, $u$tinebit. Potentia ve <marg>1 <I>Huius<*></I></marg> |
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3163 rò in A $u$tinens pondus BE vecte AB in H appen$um ad ip$um |
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3164 pondus eandem habet proportionem, quam DH ad DA; eadem |
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3165 ergo potentia in A $u$tinens pondus BE in punctis BQ $u$tenta |
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3166 tum ad ip$um pondus erit, vt DH ad DA. Similiter o$tende- |
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3167 tur pondus BE $i in G $u$tineatur, manere; $icuti à punctis BP |
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3168 $u$tinebatur: & in puncto k, vt à punctis BR. quare potentia in |
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3169 L $u$tinens pondus BE ad ip$um pondus ita erit, vt NG ad NL. |
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3170 potentia verò in M ad pondus, vt OK ad OM; hoc e$t vt di$tan |
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3171 tia à fulcimento ad punctum, vbi à centro grauitatis ponderis ho |
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3172 riz onti ducta perpendicularis vectem $ecat, ad di$tantiam à fulci- |
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3173 mento ad potentiam. quod demon$trare quoq; oportebat. |
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3174 <p>Si verò LAM e$$ent fulcimenta, & potentiæ in NDO; $imi |
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3175 liter o$tendetur ita e$$e potentiam in N ad pondus, vt LG ad L |
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3176 N; & potentiam in D, vt AH ad AD; & potentiam in O, vt |
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3177 Mk ad MO. |
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3178 <foot>M</foot> |
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3179 <pb> |
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3180 <p>Et $i vectes BA |
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3181 BL BM habeant |
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3182 fulcimenta in B, & |
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3183 pondus $upra vect&etilde; |
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3184 $it NO; & ab eius |
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3185 centro grauitatis F |
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3186 ducatur ip$i AB, & |
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3187 horizonti perpendi |
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3188 cularis FDEG; $int |
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3189 qué potentiæ in L |
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3190 AM; $imiliter o- |
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3191 $tendetur ita e$$e po |
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3192 tentiam in L pon- |
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|
3193 <fig> |
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3194 dus $u$tinentem ad ip$um pondus, vt BD ad BL; & potentiam |
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3195 in A ad pondus, vt BE ad BA, atq; potentiam in M, vt BG |
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3196 ad BM. |
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3197 <p>Sit deniq; |
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3198 vectis AB ho |
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3199 rizonti æqui- |
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|
3200 di$tans, cuius |
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|
3201 fulcimentum |
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3202 C, & pondus |
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|
3203 DE habeat c&etilde; |
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3204 trum grauita- |
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|
3205 tis F in ip $o |
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3206 vecte AB; |
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3207 $intq; deniq; |
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|
3208 alii vectes G |
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3209 H kL, quo- |
|
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|
3210 <fig> |
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3211 rum fulcimenta $int MN; pondusq; in vecte GH $u$tineatur à |
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3212 punctis GO; in vecte autem AB à punctis AP; & in uecte KL |
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3213 à punctis KQ; & centrum grauitatis F $it quoq; in utroq; uecte |
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3214 GH kL; $intq; potentiæ in HBL. Dico potentiam in H ad |
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3215 pondus ita e$$e, ut NF ad NH; & potentiam in B ad pondus, ut |
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3216 CF ad CB; ac potentiam in L ad pondus, ut MF ad ML. Quo- |
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3217 niam enim F centrum e$t grauitatis ponderis DE, $i igitur in F |
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|
3218 <pb n=46> |
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3219 $u$tineatur, pondus DE manebit $icut prius, per deffinitionem cen |
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3220 tri grauitatis; eritq; ac$iin F e$$et appen$um; atq; in vecte eodem |
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3221 modo manebit, $iue à punctis AP, $iue à puncto F $u$tineatur. |
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3222 quod idem in vectibus GH kL eueniet; $cilicet pondus eodem mo |
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3223 do manere, $iue in F, $iue in GO, vel in kQ $u$tineatur. eadem |
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3224 igitur potentia in B idem pondus DE, vel in F, vel in AP appen$um |
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3225 $u$tinebit: & quando appen$um e$t in F ad ip$um pon- |
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|
3226 dus e$t, vt CF ad CB, ergo potentia $u$tinens pondus DE in |
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3227 AP appen$um ad ip$um pondus erit, vt CF ad CB. eodemq; mo |
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3228 do potentia in H ad pondus in GO appen$um ita erit, vt NF ad |
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3229 NH. potentiaq; in L ad pondus in kQ appen$um erit, vt MF |
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3230 ad ML. quod o$tendere quoq; oportebat. |
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3231 <p>Si verò HBL e$$ent ful<*>imenta, & potentiæ e$$ent in NCM; $i- |
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|
3232 militer o$tendetur potentiam in N ad pondus ita e$$e, vt HF ad |
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3233 HN; & potentiam in C, vt BF ad BC, & potentiam in M, vt |
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|
3234 LF ad LM. |
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|
3235 <p>Et $i vectes BA |
|
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|
3236 BC BD habeãt ful |
|
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|
3237 cimenta in B, $intq; |
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|
3238 pondera in EF GH |
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|
3239 kL, ita vt eorum |
|
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|
3240 centra MNO gra- |
|
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|
3241 uitatis $int in vecti |
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|
3242 bus; $intq; poten- |
|
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|
3243 tiæ in CAD: $imi |
|
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|
3244 liter o$tendetur po |
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|
3245 tentiam in C ad |
|
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|
3246 pondus EF ita e$$e, |
|
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|
3247 <fig> |
|
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3248 vt BM ad BC, & potentiam in A ad pondus GH, vt BN ad |
|
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3249 BA, potentiamq; in D ad pondus KL, vt BO ad BD. |
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|
3250 <foot>M 2</foot> |
|
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|
3251 <pb> |
|
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3252 <head>PROPOSITIO VI.</head> |
|
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3253 <p>Sit AB recta linea, cui ad angulos $it rectos |
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3254 AD, quæ ex parte A producatur vtcunq; v$q; |
|
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3255 ad C; connectaturq; CB, quæ ex parte B quoq; |
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3256 producatur v$q; ad E. ducantur deinde à pun- |
|
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3257 cto B vtcunq; inter AB BE lineæ BF BG ip$i |
|
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3258 AB æquales; à puncti$q; FG ip$is perpendicula- |
|
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3259 res ducantur FH GK, quæ & inter $e $e, & ip$i |
|
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3260 AD con$tituantur æ- |
|
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3261 quales, ac $i BA AD |
|
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|
3262 motæ $int in BF FH, |
|
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|
3263 & in BG GK; con- |
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|
3264 nectanturq; CH CK, |
|
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|
3265 quæ lineas BF BG |
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|
3266 in punctis MN $e- |
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|
3267 cent. Dico BN mi- |
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|
3268 norem e$$e BM, & |
|
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|
3269 BM ip$a BA. |
|
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|
3270 <fig> |
|
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|
3271 <p>Connectantur BD BH |
|
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|
3272 BK. & quoniam duæ lineæ |
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|
3273 DA AB duabus HF FB |
|
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|
3274 $unt æquales, & angulus |
|
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|
3275 DAB rectus recto HFB e$t |
|
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3276 <marg>4 <I>Primi.</I></marg> etiam æqualis; erunt reliqui |
|
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|
3277 anguli reliquis angulis æqua- |
|
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|
3278 les, & HB ip$i DB æqualis. |
|
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|
3279 fimiliter o$tendetur triangu- |
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|
3280 lum BkG triangulo BHF æqualem e$$e. quare centro B, inter- |
|
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|
3281 <pb n=47> |
|
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|
3282 uallo quidem vna ip$arum circulus de$cribatur DH kE, qui li- |
|
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|
3283 neas CH CK $ecet in punctis OP; connectanturq; OB PB. |
|
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|
3284 Quoniam igitur punctum k propius e$t ip$i E, quàm H; erit linea <marg>8 <I>Tertii.</I></marg> |
|
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3285 Ck maior ip$a CH, & CP ip$a CO minor: ergo PK ip$a OH |
|
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|
3286 maior erit. Quoniam autem triangulum BkP æquicrure latera |
|
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|
3287 Bk BP lateribus BH BO trianguli BHO æquicruris æqualia ha |
|
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|
3288 bet, ba$im verò KP ba$i HO maiorem, erit angulus kBP an- <marg>25 <I>Primi.</I></marg> |
|
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|
3289 gulo HBO maior. ergo reliqui ad ba$im anguli, hoc e$t kPB |
|
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|
3290 PkB $imul $umpti, qui inter $e $unt æquales, reliquis ad ba$im an- |
|
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|
3291 gulis, nempè OHB HOB, qui etiam inter $e $unt æquales, mino- <marg>5 <I>Primi.</I></marg> |
|
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|
3292 res erunt: cùm omnes anguli cuiu$cunq; trianguli duobus $int rectis |
|
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|
3293 æquales. quare & horum dimidii, $cilicet NkB minor MHB. |
|
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|
3294 Cùm autem angulus BkG æqualis $it angulo BHF, erit NkG |
|
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|
3295 ip$o MHF maior. $i igitur à puncto k con$tituatur angulus GKQ |
|
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|
3296 ip$i FHM æqualis, fiet triangulum GkQ triangulo FHM æqua |
|
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|
3297 le; nam duo anguli ad FH vnius duobus ad Gk alterius $unt |
|
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|
3298 æquales, & latus FH lateri Gk e$t æquale, erit GQ ip$i FM æ- <marg>26 <I>Primi.</I></marg> |
|
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|
3299 quale. ergo GN maiorerit ip$a FM. Cùm itaq; BG ip$i BF $it æqua |
|
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|
3300 lis, erit BN minor ip$a BM. Quòd autem BM $it ip$a BA minor, |
|
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|
3301 e$t manife$tum; cùm BM ip$a BF, quæ ip$i BA e$t æqualis, $it |
|
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|
3302 minor. quod demon$trare oportebat. |
|
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|
3303 <p>In$uper $i intra BG BE alia vtcunq; ducatur linea ip$i BG æ- |
|
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|
3304 qualis; fiatq; operatio, quemadmodum $upra dictum e$t; $imili- |
|
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|
3305 ter o$tendetur lineam BR minorem e$$e BN. & quò propius fue |
|
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|
3306 rit ip$i BE, adhuc minorem $emper e$$e. |
|
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|
3307 <pb> |
|
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|
3308 <p>Si verò æqualia triangula BFH BGK $int |
|
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|
3309 deor$um inter BC BA con$tituta; connectan- |
|
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|
3310 turq; HC KC, quæ lineas BF BG ex parte |
|
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|
3311 FG productas in punctis MN $ecent erit BN |
|
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|
3312 maior BM, & BM ip$a BA. |
|
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|
3313 <p>Nam producatur CH |
|
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|
3314 Ck v$q; ad circumfe rentiam |
|
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|
3315 in OP, Connectanture; BO |
|
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|
3316 BP; $imili modo o$tende- |
|
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|
3317 tur lineam Pk maiorem e$ |
|
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|
3318 $e OH, angulumq; PkB mi |
|
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|
3319 norem e$$e angulo OHB. & |
|
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|
3320 quoniam angulus BHF e$t |
|
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|
3321 æqualis angulo BkG; erit to |
|
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diff
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|
3322 tus PKG angulus angulo |
|
22d6a63640c6
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casties
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diff
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|
3323 OHF minor: quare reliquus |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
3324 GKN reliquo FHM maior |
|
22d6a63640c6
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|
3325 erit. $i it aq; con$tituatur angu |
|
22d6a63640c6
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|
3326 lus GkQ ip$i FHM æqua |
|
22d6a63640c6
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|
3327 lis, linea KQ ip$am GN ita |
|
22d6a63640c6
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|
3328 $ecabit, vt GQ ip$i FM æqua |
|
22d6a63640c6
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|
3329 lis euadat: quare maior. erit |
|
22d6a63640c6
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|
3330 GN, quàm FM; quibus $i |
|
22d6a63640c6
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|
3331 æquales adiiciantur BF BG, |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
3332 erit BN ip$a BM maior. & |
|
22d6a63640c6
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|
3333 cùm BM $it ip$a FB maior, |
|
22d6a63640c6
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|
3334 erit quoq; ip$a BA maior. $i |
|
22d6a63640c6
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|
3335 militer o$tendetur, quò pro |
|
22d6a63640c6
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casties
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diff
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|
3336 pius fuerit BG ip$i BC, li- |
|
22d6a63640c6
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|
3337 neam BN $emper maiorem |
|
22d6a63640c6
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|
3338 e$$e. |
|
22d6a63640c6
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|
3339 <fig> |
|
22d6a63640c6
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|
3340 <pb n=48> |
|
22d6a63640c6
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|
3341 <head>PROPOSITIO VII.</head> |
|
22d6a63640c6
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|
3342 <p>Sit recta linea AB, cuì perpendi cularis exi- |
|
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|
3343 $tat AD, quæ ex parte D producatur vtcunq; v$q; |
|
22d6a63640c6
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|
3344 ad C; connectaturq; CB, quæ producatur e- |
|
22d6a63640c6
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|
3345 tiam v$q, ad E; & inter AB BE lineæ $imiliter |
|
22d6a63640c6
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|
3346 vtcunq; ducantur BF BG ip$i AB æquales; à |
|
22d6a63640c6
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|
3347 punctisq; FG lineæ FH GK ip$i AB æquales, |
|
22d6a63640c6
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|
3348 ip$is verò BF BG per- |
|
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|
3349 p&etilde;diculares ducantur; |
|
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|
3350 ac $i BA AD motæ |
|
22d6a63640c6
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|
3351 $int in BF FH BG |
|
22d6a63640c6
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|
3352 GK: Connectanturq; |
|
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|
3353 CH CK, quæ lineas |
|
22d6a63640c6
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|
3354 BF BG productas $e- |
|
22d6a63640c6
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|
3355 cent in punctis MN. |
|
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|
3356 Dico BN maiorem e$ |
|
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|
3357 $e BM, & BM ip$a BA. |
|
22d6a63640c6
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|
3358 <fig> |
|
22d6a63640c6
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|
3359 <p>Connectantur BD BH Bk, |
|
22d6a63640c6
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|
3360 & centro B, interuallo quidem |
|
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|
3361 BD, circulus de$cribatur. $imi |
|
22d6a63640c6
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|
3362 liter vt in præcedenti demon- |
|
22d6a63640c6
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|
3363 $trabimus puncta kHDOP in |
|
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|
3364 circuli circumferentia e$$e, trian |
|
22d6a63640c6
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|
3365 gulaq; ABD FBH GBk in- |
|
22d6a63640c6
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|
3366 ter $e $e æqualia e$$e, atq; lineam |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
3367 Pk maiorem OH, angulumq; |
|
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|
3368 PKB minorem e$$e angulo O |
|
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|
3369 HB. Quoniam igitur angulus BHF æqualis e$t angulo BkG, |
|
22d6a63640c6
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|
3370 <pb> |
|
22d6a63640c6
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parents:
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changeset
|
3371 erit totus angulus PkG angu- |
|
22d6a63640c6
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|
3372 lo OHF minor: quare reliquus |
|
22d6a63640c6
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|
3373 GkN reliquo FHM maior |
|
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|
3374 erit. $i igitur fiat angulus GK |
|
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|
3375 Q ip$i FHM æqualis, erit trian |
|
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parents:
diff
changeset
|
3376 gulum GKQ triangulo FHM |
|
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|
3377 æquale, & latus GQ lateri FM |
|
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|
3378 æquale; ergo maior erit GN ip |
|
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changeset
|
3379 $a FM; ac propterea BN ma- |
|
22d6a63640c6
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|
3380 ior erit BM. BM autem ma- |
|
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|
3381 ior erit BA; nam BM maior e$t |
|
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diff
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|
3382 ip$a BF. quod demon$trare |
|
22d6a63640c6
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parents:
diff
changeset
|
3383 oportebat. |
|
22d6a63640c6
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parents:
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|
3384 <fig> |
|
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|
3385 <p>Eodemq; pror$us modo, quo |
|
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diff
changeset
|
3386 propius fuerit BG ip$i BE, li- |
|
22d6a63640c6
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parents:
diff
changeset
|
3387 neam BN $emper maiorem e$$e |
|
22d6a63640c6
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parents:
diff
changeset
|
3388 o$tendetur. |
|
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casties
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changeset
|
3389 <p>Si autem triangula BFH BGK deor$um in- |
|
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diff
changeset
|
3390 ter AB BC con$tituantur, ducanturq; CHO |
|
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diff
changeset
|
3391 CKP, quæ lineas BF BG $ecent in punctis M |
|
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|
3392 N; erit linea BN minor ip$a BM, & BM |
|
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|
3393 ip$a BA. |
|
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parents:
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changeset
|
3394 <pb n=49> |
|
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parents:
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changeset
|
3395 <p>Connectantur enim BO BP, |
|
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parents:
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changeset
|
3396 $imiliter o$tendetur angulum |
|
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changeset
|
3397 PKB minorem e$$e OHB. & |
|
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parents:
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changeset
|
3398 quoniam angulus FHB æqua- |
|
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parents:
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changeset
|
3399 lis e$t angulo GkB; erit angu |
|
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changeset
|
3400 lus GkN angulo FHM ma- |
|
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parents:
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changeset
|
3401 ior: quare & linea GN ma- |
|
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parents:
diff
changeset
|
3402 ior erit ip$a FM. ideoq; linea |
|
22d6a63640c6
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parents:
diff
changeset
|
3403 nea BN minor erit linea BM. |
|
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changeset
|
3404 Cùm autem maior $it BF ip$a |
|
22d6a63640c6
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changeset
|
3405 BM; erit BM ip$a BA minor. Si- |
|
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parents:
diff
changeset
|
3406 miliq; modo o$tendetur, quò |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
3407 propius fuerit BG ip$i BC, li- |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
3408 neam BN $emper minorem |
|
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casties
parents:
diff
changeset
|
3409 e$$e. |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
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|
3410 <fig> |
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3411 <head>PROPOSITIO VIII.</head> |
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3412 <p>Potentia pondus $u$tinens centrum grauitatis |
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3413 $upra vectem horizonti æquidi$tantem habens, |
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3414 quò magis pondus ab hoc $itu vecte eleuabitur; |
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3415 minori $emper, vt $u$tineatur, egebit potentia: |
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3416 $i verò deprimetur, maiori. |
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3417 <foot>N</foot> |
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3418 <pb> |
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3419 <fig> |
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3420 <p>Sit vectis AB horizonti æquidi$tans, cuius fulcimentum C; |
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3421 pondus autem BD, eiu$dem verò grauitatis centrum $it $upra ve |
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3422 ctem vbi H: $itq; potentia $u$tinens in A. moueatur deinde ve |
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3423 ctis AB in EF, $itq; pondus motum in FG. Dico primùm mino |
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3424 rem potentiã in E $u$tinere pondus FG vecte EF, quàm pot&etilde;tia in |
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3425 A pondus BD vecte AB. $it k centrum grauitatis ponderis FG; |
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3426 deinde tùm ex H, tùm ex K ducantur HL kM ip$orum horizon |
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3427 tibus perpendiculares, quæ in centrũ mundi conuenient; $itq; HL ip |
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3428 $i quoq; AB perpendicularis. ducatur deinde kN ip$i EF perpen- |
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3429 dicularis, quæ ip$i HL æqualis erit, & CN ip$i CL æqualis. Quo- |
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3430 <marg>5 <I>Huius.</I></marg> niam enim HL horizonti e$t perpendicularis, potentia in A $u |
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3431 $tinens pondus BD ad ip$um pondus eam habebit proportionem, |
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3432 quam CL ad CA. rur$us quoniam kM horizonti e$t perpendicu |
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3433 laris, potentia in E pondus FG $u$tinens ita erit ad pondus, vt |
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3434 CM ad CE. Cùm autem CN NK ip$is CL LH $int æquales, |
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3435 <marg>6 <I>Huius.</I></marg> angulosq; rectos contineant; erit CM minor ip$a CL; ergo CM |
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3436 <marg>8 <I>Quinti.</I></marg> ad CA minorem habebit proportionem, quam CL ad CA; & |
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3437 <pb n=45> |
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3438 CA ip$i CE e$t æqualis, minorem igitur proportionem habebit |
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3439 CM ad CE. quàm CL ad CA: & cùm pondera BD FG $int |
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3440 æqualia, e$t enim idem pondus; ergo minor erit proportio po |
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3441 tentiæ in E pondus FG $u$tinentis ad ip$um pondus, quàm po |
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3442 tentiæ in A pondus BD $u$tinentis ad ip$um pondus. Quare <marg>10 <I>Quinti.</I></marg> |
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3443 minor potentia in E $u$tinebit pondus FG, quàm potentia in A |
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3444 pondus BD. & quò pondus magis eleuabitur; $emper o$tendetur |
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3445 minorem adhuc potentiam pondus $u$tinere; cùm linea PC mi <marg>6 <I>Huius.</I></marg> |
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3446 nor $it linea CM. $it deinde vectis in QR, & pondus in QS, |
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3447 cuius centrũ grauitatis $it O. dico maiorem requiri potentiam in R |
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3448 ad $u$tinendũ pondus QS, quàm in A ad pondus BD. ducatur à cen |
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3449 tro grauitatis O linea OT horizonti perpendicularis. & quo- |
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3450 niam HL OT, $i ex parte L, atq; T producantur, in centrum |
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3451 mundi conuenient; erit CT maior CL: e$t autem CA ip$i CR <marg>6 <I>Huius.</I></marg> |
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3452 æqualis, habebitergo TC ad CR maiorem proportionem, quàm |
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3453 LC ad CA. Maior igitur erit potentia in R $u$tinens pondus <marg>8 <I>Quinti. Ex</I> 10 <I>quin ti.</I></marg> |
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3454 QS, quàm in A $u$tinens BD. $imiliter o$tendetur; quò vectis |
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3455 RQ magis à vecte AB di$tahit deor$um vergens, $emper maio- |
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3456 rem potentiam requiri ad $u$tinendum pondus: di$tantia enim CV <marg>6 <I>Huius.</I></marg> |
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3457 longior e$t CT. Quò igitur pondus à $itu horizonti æquidi$tan |
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3458 temagis eleuabitur à minori $emper potentia pondus $u$tinebitur; |
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3459 quò verò magis deprimetur, maiori, vt $u$tineatur, egebit potentia. |
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|
3460 quod demon$trare oportebat. |
|
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3461 <p>Hinc facile elicitur potentiam in A ad poten- |
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|
3462 tiam in E ita e$$e, vt CL ad CM. |
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3463 <p>Nam ita e$t LC ad CA, vt potentia in A ad pondus; vt au- |
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3464 tem CA, hoc e$t CE ad CM, ita e$t pondus ad potentiam in E; |
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3465 quare ex æquali potentia in A ad potentiam in E ita erit, vt CL <marg>22 <I>Quinti.</I></marg> |
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|
3466 ad CM. |
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3467 <p>Similiq; ratione non $olum o$tendetur, potentiam in A ad po- |
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3468 tentiam in Rita e$$e, vt CL ad CT; $ed & potentiam quoq; in E |
|
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3469 ad potentiam in R ita e$$e, vt CM ad CT. & ita in reliquis. |
|
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|
3470 <foot>N 2</foot> |
|
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|
3471 <pb> |
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|
3472 <fig> |
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3473 <p>Sit deinde vectis AB horizonti æquidi$tans, cuius fulcimen- |
|
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|
3474 tum B; & centrum grauitatis H ponderis CD $it $upra vectem; |
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3475 moueaturq; vectis in BE, pondu$q; in FG. dico minorem po- |
|
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3476 tentiam in E $u$tinere pondus FG vecte EB, quàm potentia in |
|
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|
3477 A pondus CD vecte AB. $it k centrum grauitatis ponderis FG, |
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3478 & à centris grauitatum Hk ip$orum horizontibus perpendicu- |
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|
3479 <marg>6 <I>Huius.</I></marg> lares ducantur HL kM. Quoniam enim (ex $upra demon$tratis) |
|
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3480 <marg>8 <I>Quinti.</I></marg> BM minor e$t BL, & BE ip$i BA æqualis; minorem habebit |
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3481 <marg>5 <I>Huius.</I></marg> proportionem BM ad BE, quàm BL ad BA. $ed vt BM ad |
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3482 BE, ita potentia in E $u$tinens pondus FG ad ip$um pondus; & |
|
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3483 vt BL ad BA, ita potentia in A ad pondus CD; minorem |
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|
3484 habebit proportionem potentia in E ad pdndus FG, quàm poten |
|
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3485 <marg>10 <I>Quinti.</I></marg> tia in A ad pondus CD. Ergo potentia in E minor erit poten- |
|
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3486 tia in A. $imiliter o$tendetur, quò magis pondus eleuabitur, $em- |
|
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|
3487 per minorem potentiam pondus $u$tinere. Sit autem vectis in |
|
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3488 BO, & pondus in PQ, cuius cenrtum grauitatis $it R. dico maio |
|
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|
3489 rem potentiam in O requiri ad $u$tinendum pondus PQ vecte BO, |
|
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|
3490 quàm pondus CD vecte BA. ducatur à puncto R horizonti per- |
|
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|
3491 <marg>6 <I>Huius.</I></marg> pendicularis RS. & quoniam BS maior e$t BL, habebit BS ad |
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|
3492 BO maiorem proportionem, quàm BL ad BA; quare maior erit |
|
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|
3493 potentia in O $u$tinens pondus PQ, quàm potentia in A $u$ti |
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|
3494 nens pondus CD. & hoc modo o$tendetur' quò vectis BO ma |
|
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3495 gis à vecte AB deor$um tendens di$tabit, $emper maiorem ponderi |
|
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|
3496 <pb n=51> |
|
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|
3497 $u$tinendo requiri potentiam. |
|
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|
3498 <p>Hinc quoq; vt $upra patet pontentiam in A ad potentiam in E e$ |
|
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|
3499 $e, vt BL ad BM; potentiamq; in A ad potentiam in O, vt BL |
|
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|
3500 ad BS. atque potentiam in E ad potentiam in O, vt BM |
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|
3501 ad BS. |
|
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3502 <p>Præterea $i in B alia intelligatur potentia, ita vt duæ $int poten |
|
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3503 tiæ pondus $u$tinentes; minor erit potentia in B $u$tinens pon- |
|
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3504 dus PQ vecte BO, quàm pondus CD vecte BA. ex aduer$o au |
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|
3505 tem maior requiritur potentia in B ad $u$tinendum pondus FG ve |
|
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|
3506 cte BE, quàm pondus CD vecte AB. ducta enim kN ip$i EB |
|
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3507 perpendicularis, erit EN ip$i AL æqualis: quare EM ip$a LA |
|
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|
3508 maior erit. ergo maiorem habebit proportionem EM ad E<I>B</I>, <marg>8 <I>Quinti.</I></marg> |
|
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|
3509 quàm LA ad A<I>B</I>; & LA ad A<I>B</I> maiorem, quàm SO ad O<I>B</I>; <marg>5 <I>Huius.</I></marg> |
|
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3510 quæ $unt proportiones potentiæ ad pondus. |
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3511 <p>Similiter o$tendetur potentiam in <I>B</I> pondus vecte A<I>B</I> $u$ti- |
|
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3512 nentem ad potentiam in eodem puncto <I>B</I> vecte E<I>B</I> $u$tinentem |
|
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3513 e$$e, vt LA ad EM; ad potentiam autem in B pondus vecte O<I>B</I> |
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3514 $u$tinentem ita e$$e, vt AL ad OS. quæ verò vectibus E<I>B</I> OB |
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3515 $u$tinent inter $e $e e$$e, vt EM ad OS. |
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3516 <p>Deinde vt in iis, quæ $uperius dicta $unt, demon$trabimus po- |
|
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|
3517 tentiam in <I>B</I> ad potentiam in E eam habere proportionem, quam <marg>3 <I>Cor.</I></marg> |
|
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|
3518 EM ad M<I>B</I>; & potentiam in <I>B</I> ad potentiam in A ita e$$e, vt AL ad <marg>2 <I>Huius.</I></marg> |
|
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|
3519 L<I>B</I>, potentiamq; in <I>B</I> ad potentiam in O, vt OS ad S<I>B.</I> |
|
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3520 <p>Sit autem vectis A<I>B</I> |
|
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|
3521 horizonti æquidi$tans, |
|
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|
3522 cuius fulcimentum <I>B</I>, |
|
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|
3523 grauitati$q; centrum H |
|
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|
3524 ponderis AC $it $upra |
|
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|
3525 vectem: moueaturq; ve |
|
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|
3526 ctis in <I>B</I>E, ac pondus |
|
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|
3527 in EF, potentiaq; in G. |
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|
3528 $imiliter vt $upra o$ten- |
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|
3529 detur potentiam in G |
|
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|
3530 pondus EF $ui$tinen- |
|
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|
3531 <fig> |
|
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|
3532 tem minorem e$$e potentia in D pondus AC $u$tinente. cùm |
|
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|
3533 <pb> |
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|
3534 enim minor $it BM ip$a |
|
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|
3535 BL, minorem habebit |
|
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|
3536 proportionem MB ad |
|
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|
3537 BG, quàm LB ad BD. |
|
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|
3538 atq; hoc modo o$ten- |
|
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|
3539 detur, quò pondus ve- |
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|
3540 cte magis eleuabitur, mi |
|
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|
3541 norem $emper. ad pon- |
|
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|
3542 dus $u$tinendum requi- |
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|
3543 ri potentiam. Simili- |
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|
3544 ter $i moucatur vectis |
|
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|
3545 in BO, potentiaq; $u- |
|
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|
3546 <fig> |
|
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|
3547 $tinens in N, o$tendetur potentiam in N maiorem e$$e potentia in |
|
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|
3548 D. maiorem enim habet proportionem SB ad BN, quàm LB |
|
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|
3549 ad BD. o$tendetur etiam, quò magis pondus deprimetur; ma- |
|
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|
3550 iorem $emper (vt $u$tineatur) requiri potentiam. quod demon |
|
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|
3551 $trare oportebat. |
|
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|
3552 <p>Hinc quoq; liquet potentias in GDN inter $e $e ita e$$e, vt |
|
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|
3553 BM ad BL, atq; vt BL ad BS, deniq; vt BM ad BS. |
|
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|
3554 <head>COROLLARIVM.</head> |
|
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|
3555 <p>Ex his manife$tum e$t; $i potentia vecte $ur- |
|
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|
3556 $um moueat pondus, cuius centrum grauitatis |
|
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|
3557 $it $upra vectem, quò magis pondus eleuabitur; |
|
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|
3558 $emper minorem potentiam requiri vt pondus |
|
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|
3559 moueatur. |
|
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|
3560 <p>Vbi enim potentia pondus $u$tinens e$t $emper minor, erit |
|
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|
3561 quoq; potentia ip$um mouens $emper minor. |
|
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|
3562 <pb n=52> |
|
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|
3563 <fig> |
|
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|
3564 <p>Ex iis etiam demon$trabitur, $i centrum grauitatis eiu$dem pon |
|
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|
3565 deris, $iue propinquius, $iue remotius fuerit à vecte AB horizon- |
|
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|
3566 ti æquidi$tante, eandem potentiam in A pondus nihilominus |
|
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|
3567 $u$tinere: vt$i centrum grauitatis H ponderis BD longius ab$it |
|
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|
3568 à vecte BA, quàm centrum grauitatis N ponderis PV, dum- |
|
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|
3569 modo ducta à puncto H perpendicularis HL horizonti, vectiq; |
|
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3570 AB tran$eat per N; $itq; pondus PV ponderi BD æquale; |
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3571 erit tùm pondus BD, tùm pondus PV, ac $i ambo in L e$- |
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3572 $ent appen$a; atque $unt æqualia, cùm loco vnius ponderis ac- |
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3573 cipiantur, eadem igitur potentia in A $u$tinens pondus BD, |
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3574 pondus quoq; PV $u$tinebit. Vecte autem EF, quò centrum |
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3575 grauitatis longius fuerit à vecte, eò facilius potentia idem pon- |
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3576 dus $u$tinebit: vt $i centrum grauitatis k ponderis FG longius |
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3577 $it à vecte EF, quàm centrum grauitatis X ponderis YZ; itata |
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3578 men vt ducta à puncto k vecti FE perpendicularis tran$eat per |
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3579 X; $itq; pondus FG ponderi YZ æquale; & à punctis kX ip- |
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3580 $o<*>um horizontibus perpendiculares ducantur KM X9; erit C9 |
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3581 maior CM; ac propterea pondus FG in vecte erit, ac $i in M e$ |
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3582 $et appen$um, & pondus YZ, ac $i in 9 e$$et appen$um. quo |
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|
3583 <pb> |
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3584 <fig> |
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3585 <marg>8 <I>Quinti.</I></marg> niam autem maiorem habet proportionem C9 ad CE, quàm |
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3586 CM ad CE, maior potentia in E fu$tinebit pondus YZ, quàm |
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3587 FG. In vecte autem QR è conuer$o demon$trabitur, $cilicet |
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3588 quò centrum grauitatis eiu$dem ponderis $it longius à vecte, eò |
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3589 maiorem e$$e potentiam pondus $u$tinentem. maior enim e$t |
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3590 CT, quàm CI; & obid maiorem habebit proportionem CT |
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3591 ad CR, quàm CI ad CR. Similiter demon$trabitur, $i pondus |
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3592 intra potentiam, & fulcimentum fuerit collocatum; vel poten- |
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3593 tia intra fulcimentum, & pondus. Quod idem etiam potentiæ |
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3594 eueniet mouenti. vbi enim minor potentia $u$tinet pondus, ibi |
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3595 minor potentia mouebit; & vbi maior in $u$tinendo, ibi maior |
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|
3596 quoq; in mouendo re quiretur. |
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|
3597 <head>RROPOSITIO VIIII.</head> |
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3598 <p>Potentia pondus $u$tinens infra vectem ho- |
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3599 rizonti æquidi$tantem ip$ius centrum grauitatis |
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|
3600 <pb n=53> |
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3601 habens, quò magis ab hoc $itu vecte pondus ele |
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3602 uabitur maiori $emper potentia, vt $u$tineatur, |
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3603 egebit. $i verò deprimetur, minori. |
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|
3604 <fig> |
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3605 <p>Sit vectis AB horizonti æquidi$tans, cuius fulcimentum C; |
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3606 $itq; pondus AD, cuius centrum grauitatis L $it infra vectem; |
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3607 $itq; potentia in B $u$tinens pondus AD: moueatur deinde ve- |
|
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3608 ctis in FG, & pondus in FH. Dico primum maiorem requiri |
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3609 potentiam in G ad $u$tinendum pondus FH vecte FG, quàm |
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3610 $it potentia in B pondere exi$tente AD vecte autem AB. $it M |
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3611 grauitatis centrum ponderis FH, & à punctis LM ip$orum ho- |
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3612 rizontibus perpendiculares ducantur Lk MN: ip$i verò FG per- |
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3613 pendicularis ducatur MS, quæ æqualis erit LK, & CK ip$i CS |
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3614 erit etiam æqualis. Quoniam igitur CN maior e$t Ck, habe- <marg>7 <I>Huius.</I></marg> |
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3615 bit NC ad CG maiorem proportionem, quàm Ck ad CB; po <marg>8 <I>Quinti.</I></marg> |
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3616 tentia uerò in B ad pondus AD eandem habet, quam kC ad CB: <marg>5 <I>Huius.</I></marg> |
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3617 & vt potentia in G ad pondus FH, ita e$t NC ad CG; ergo |
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3618 maiorem habebit proportionem potentia in G ad pondus FH, |
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3619 quàm potentia in B ad pondus AD. maior igitur e$t potentia <marg>10 <I>Quinti</I></marg> |
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3620 in Gip$a potentia in B. $i verò vectis $it in OP, & pondus in |
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3621 OQ; erit potentia in B maior, quàm in P. eodem enim mo- |
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3622 do o$tendetur CR minorem e$$e Ck, & CR ad CP minorem <marg>7 <I>Huius.</I></marg> |
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|
3623 <foot>O</foot> |
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|
3624 <pb> |
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|
3625 <fig> |
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3626 habere proportionem, quàm Ck ad CB; & ob id potentiam in |
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3627 B maiorem e$$e potentia in P. & hoc modo o$tendetur, quò ma- |
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3628 gis à $itu AB pondus eleuabitur, $emper maiorem potentiam ad |
|
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3629 pondus $u$tinendum requiri. è contra verò $i deprimetur. quod |
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|
3630 demon$trare oportebat. |
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3631 <p>Hinc quoq; facilè elici pote$t potentias in PBG inter $e $e ita |
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3632 e$$e, vt CR ad Ck; & vt Ck ad CN; atq; vt CN ad CR. |
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|
3633 <fig> |
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3634 <p>Sit deinde vectis AB horizonti æquidi$tans, cuius fulcimentum |
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|
3635 B; pondu$q; CD habeat centrum grauitatis O infra vectem; $itq; |
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|
3636 potentia in A $u$tinens pondus CD. Moueatur deinde vectis in |
|
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|
3637 <pb n=54> |
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3638 BE BF, pondu$q; transferatur in GH kL. Dico maiorem re- |
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3639 quiri potentiam in E, vt pondus $u$tineatur, quàm in A; & ma |
|
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3640 iorem in A, quàm in F. ducantur à centris grauitatum horizon- |
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3641 tibus perpendiculares NM OP QR, quæ ex parte NOQ |
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3642 protractæ in centrum mundi conuenient. $imiliter vt $upra o$ten |
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|
3643 detur BM maior&etilde; e$$e BP, & <I>B</I>P maiorem BR; & BM ad BE ma- <marg>7 <I>Huius.</I></marg> |
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3644 iorem habere proportionem, qaàm BP ad BA; & BP ad BA ma- |
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3645 iorem, quàm BR ad BF: & propter hoc potentiam in E maio- |
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|
3646 rem e$$e potentia in A; & potentiam in A maiorem potentia in |
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3647 F. & quò vectis magis à $itu AB eleuabitur, $emper o$tendetur, |
|
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3648 maiorem requiri potentiam ponderi $u$tinendo. $i verò depri- |
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|
3649 metur, minorem. |
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|
3650 <p>Hinc patet etiam potentias in EAF inter $e $e ita e$$e, vt BM ad |
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|
3651 BP; & vt BP ad BR; ac vt BM ad BR. |
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3652 <p>In$uper $i in B altera $it potentia, ita vt duæ $int potentiæ pondus |
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3653 $u$tinentes, maiore opus e$t potentia in B pondus kL $u$tinente |
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|
3654 vecte BF, quàm pondus CD vecte AB. & adhuc maiore vecte |
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3655 AB, quàm vecte BE. maiorem enim habet proportionem RF |
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3656 ad FB, quàm PA ad AB; & PA ad AB maiorem habet, quàm |
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|
3657 EM ad EB. |
|
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3658 <p>Similiterq; o$tendetur potentias in B pondus vectibus $u$tinen- |
|
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3659 tes inter $e $e ita e$$e, vt EM ad AP; & ut |
|
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|
3660 AP ad FR; atque ut |
|
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|
3661 EM ad FR. |
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|
3662 <p>Præterea potentia in Bad potentiam in F ita erit, ut RF ad <marg>3 <I>Cor.</I></marg> |
|
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3663 RB; & potentia in B ad potentiam in A, ut PA ad PB, & po- <marg>2 <I>Huius.</I></marg> |
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3664 tentia in <I>B</I> ad potentiam in E, ut EM ad M<I>B.</I> |
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|
3665 <foot>O 2</foot> |
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|
3666 <pb> |
|
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3667 <p>Sit autem vectis |
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|
3668 AB horizonti æqui- |
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3669 di$tans, cuius fulci- |
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|
3670 mentum B; & pon- |
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|
3671 dus AC, cuius cen- |
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|
3672 trum grauitatis $it in- |
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|
3673 fra vectem: $itq; po- |
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|
3674 tentia in D pondus |
|
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|
3675 $u$tin&etilde;s; moueaturq; |
|
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|
3676 vectis in BE BF, & |
|
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|
3677 potentia in GH: $i- |
|
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|
3678 militer o$tendetur po |
|
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|
3679 <fig> |
|
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|
3680 tentiam in G maiorem e$$e debere potentia in D; & potentiam in |
|
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|
3681 D maiorem potentia in H. maiorem enim proportionem habet |
|
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|
3682 KB ad BG, quàm BL ad BD; & BL ad BD maiorem, quàm |
|
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|
3683 MB ad BH. & hoc modo o$tendetur, quò vectis magis à $itu |
|
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|
3684 AB eleuabitur, adhuc $emper maiorem e$$e debere potentiam pon |
|
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|
3685 dus $u$tinentem. quò autem magis deprimetur; minorem. quod |
|
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|
3686 demon$trare oportebat. |
|
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|
3687 <p>Similiter in his potentiæ in GDH inter $e $e ita. erunt, vt BK |
|
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|
3688 ad BL; & vt BL ad BM; deniq; vt Bk ad BM. |
|
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|
3689 <head>COROLLARIVM.</head> |
|
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|
3690 <p>Ex his patet etiam, $i potentia vecte $ur$um |
|
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|
3691 moueat pondus, cuius centrum grauitatis $it in- |
|
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|
3692 fra vectem; quò magis pondus eleuabitur, $em |
|
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|
3693 per maiorem requiri potentiam, vt pondus mo |
|
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|
3694 ueatur. |
|
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|
3695 <p>Nam $i potentia pondus $u$tinens $emper e$t maior: erit quoq; |
|
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|
3696 potentia mouens $emper maior. |
|
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|
3697 <pb n=55> |
|
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|
3698 <fig> |
|
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|
3699 <p>Et his etiam facilè elicietur, $i centrum grauitatis eiu$dem pon- |
|
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|
3700 deris, $iue propius, $iue remotius fuerit à vecte AB horizonti æ- |
|
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|
3701 quidi$tante; eandem potentiam in B pondus $u$tinere. vt $i cen- |
|
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|
3702 trum grauit atis L ponderis AD $it remotius à vecte BA, quàm |
|
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|
3703 centrum grauitatis N ponderis PV; dummodo ducta à puncto L |
|
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|
3704 perpendicularis LK horizonti, vectiq; AB tran$eat per N: $imili- |
|
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|
3705 ter vt in præcedenti o$tendetur, eandem potentiam in B, & pondus |
|
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|
3706 AD, & pondus PV $u$tinere. In vecte auté EF, quò centrũ grauitatis |
|
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|
3707 longius aberit à vecte, eò maiori opus erit potentia ponderi $u$ti- |
|
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|
3708 nendo. vt centrum grauitatis M ponderis FH remotius $it à ue |
|
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|
3709 cte EF, quàm S centrum grauitatis ponderis XZ; ducantur à pun |
|
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|
3710 ctis MS horizontibus perpendiculares MI SG; erit CI maior |
|
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|
3711 CG: ac propterea maior e$$e debet potentia in E pondus FH $u |
|
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|
3712 $tinens, quàm pondus XZ. Contra uerò in uecte OR o$tende |
|
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|
3713 tur, quò $cilicet centrum grauitatis eiu$dem ponderis longius ab |
|
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|
3714 $it à uecte, à minori potentia pondus $u$tineri. minor enim e$t |
|
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|
3715 CY, quàm CT. Simili quoq; modo demon$trabitur, $i pondus |
|
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|
3716 $it intra potentiam, & fulcimentum; uel potentia intra fulci- |
|
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|
3717 mentum, & pondus. Quod idem potentiæ eueniet mouenti: |
|
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|
3718 <pb> |
|
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|
3719 vbi enim minor potentia $u$tinet pondus, ibi minor potentia mo- |
|
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|
3720 uebit. & vhi maior potentia in $u$tinendo; ibi quoq; maior in mo |
|
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|
3721 uendo aderit. |
|
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|
3722 <head>PROPOSITIO X.</head> |
|
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|
3723 <p>Potentia pondus $u$tinens in ip$o vecte cen- |
|
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|
3724 trum grauitatis habens, quomodocunq; vecte |
|
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|
3725 transferatur pondus; eadem $emper, vt $u$tinea- |
|
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|
3726 tur, potentia opus erit. |
|
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|
3727 <fig> |
|
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|
3728 <p>Sit vectis AB horizonti æquidi$tàns, cuius fulcimentum C. |
|
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|
3729 E verò centrum grauitatis ponderis in ip$o $it vecte. Moueatur |
|
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|
3730 deinde uectis in FG, Hk; & centrum grauitatis in LM. dico ean |
|
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|
3731 dem potentiam in kBG idemmet $emper $u$tinere pondus. |
|
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|
3732 Quoniam enim pondus in uecte AB perinde $e habet, ac $i e$$et |
|
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|
3733 <marg>5 <I>Huius.</I></marg> appen$um in E; & in uecte GF, ac $i e$$et appen$um in L; & in |
|
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|
3734 uecte Hk. ac $i in M e$$et appen$um; di$tantiæ uerò CL CE |
|
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|
3735 CM $unt inter $e $e æquales; nec non CK CB CG inter $e æ- |
|
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|
3736 quales; erit potentia in B ad pondus, ut CE ad CB; atque poten |
|
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|
3737 <pb n=56> |
|
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|
3738 tia in k ad pondus, ut CM ad Ck; & potentia in G ad pondus, |
|
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|
3739 vt CL ad CG. eadem igitur potentia in k<I>B</I>G idem translatum |
|
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|
3740 pondus $u$tinebit. quod demon$trare oportebat. |
|
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|
3741 <p>Similiter o$tendetur, $i pondus e$$et intra potentiam, & fulci- |
|
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|
3742 mentum; vel potentia inter fulcimentum, & pondus. quod idem |
|
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|
3743 potentiæ mouenti eueniet. |
|
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|
3744 <head>RROPOSITIO XI.</head> |
|
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|
3745 <p>Si vectis di$tantia inter fulcimentum, & poten |
|
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|
3746 tiam ad di$tantiam fulcimento, punctoq;, vbi |
|
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|
3747 à centro grauitatis ponderis horizonti ducta |
|
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|
3748 perpendicularis vectem $ecat, interiectam ma- |
|
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|
3749 iorem habuerit proportionem, quàm pondus |
|
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|
3750 ad potentiam; pondus vtiq; à potentia moue- |
|
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|
3751 bitur. |
|
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|
3752 <p>Sit véctis AB, ex |
|
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|
3753 punctoq; A $u$penda |
|
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|
3754 tur pondus C; hoc e$t |
|
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|
3755 punctum A $emper $it |
|
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|
3756 punctum, vbi perpen |
|
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|
3757 dicularis à grauitatis |
|
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|
3758 centro ponderis du- |
|
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|
3759 cta vectem $ecat; $itq; |
|
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|
3760 <fig> |
|
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|
3761 potentia in B, ac fulcimentum $it D; & DB ad DA maiorem |
|
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|
3762 habeat proportionem, quàm pondus C ad potentiam in B. Di- |
|
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|
3763 co pondus Cà potentia in B moueri. fiat vt BD ad DA, ita |
|
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|
3764 pondus E ad potentiam in B; atq; pondus E quoq; appendatur |
|
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|
3765 in A: patet potentiam in B æqueponderare ip$i E; hoc e$t pon- <marg>1 <I>Huius.</I></marg> |
|
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|
3766 dus E $u$tinere. & quoniam BD ad DA maiorem habet pro- |
|
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|
3767 portionem, quàm Cad potentiam in B; & vt BD ad DA, ita |
|
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|
3768 <pb> |
|
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|
3769 e$t pondus E ad po- |
|
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|
3770 tentiam: igitur E ad |
|
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|
3771 potentiam maiorem |
|
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|
3772 habebit proportio- |
|
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|
3773 nem, quàm pondus |
|
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|
3774 C ad eandem poten- |
|
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|
3775 <marg>10 <I>Quinti.</I></marg> tiam. quare pondus |
|
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|
3776 E maius erit ponde- |
|
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|
3777 <fig> |
|
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|
3778 re C. & cùm potentia ip$<*> E æqueponderet, potentia igitur ip$i |
|
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|
3779 C non æqueponderabit, $ed $ua ui deor$um verget. pondus igitur |
|
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|
3780 C à potentia in B mouebitur vecte AB, cuius fulcimentum |
|
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|
3781 e$t D. |
|
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|
3782 <p>Si verò $it vectis AB, & |
|
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|
3783 fulcimentum A, pondu$q; C |
|
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|
3784 in D appen$um, & potentia |
|
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|
3785 in B; & BA ad AD maio- |
|
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|
3786 rem habeat proportionem, |
|
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|
3787 quàm pondus C ad poten- |
|
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|
3788 tiam in B. dico pondus C à |
|
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|
3789 <fig> |
|
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|
3790 potentia in B moueri. fiat vt BA ad AD; ita pondus E ad poten |
|
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|
3791 <marg>2 <I>Huius.</I></marg> tiam in B: & $i E appendatur in D, potentia in B pondus E $u$ti |
|
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|
3792 nebit. $ed cùm BA ad AD maiorem habeat proportionem, |
|
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|
3793 quàm pondus C ad potentiam in B; & vt BA ad AD, ita e$t |
|
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|
3794 pondus E ad potentiam in B: pondus igitur E ad potentiam, |
|
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|
3795 quæ e$t in B, maiorem habebit proportionem, quàm pondus C |
|
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|
3796 <marg>10 <I>Quinti.</I></marg> ad eandem potentiam. & ideo pondus E maius erit pondere C. |
|
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|
3797 potentia verò in B $u$tinet pondus E; ergo potentia in B pondus |
|
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|
3798 C minus pondere E in D appen$um mouebit vecte AB, cuius fulci |
|
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|
3799 mentum e$t A. |
|
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|
3800 <pb n=57> |
|
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|
3801 <p>Sit rur$us vectis |
|
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|
3802 AB, cuius fulcimen |
|
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|
3803 tũ A; & pondus C in |
|
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|
3804 B $it appen$um; $itq; |
|
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|
3805 potentia in D: & |
|
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|
3806 DA ad AB maio- |
|
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|
3807 rem habeat propor- |
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|
3808 tionem, quàm pon- |
|
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|
3809 <fig> |
|
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|
3810 dus C ad potentiam, quæ e$t in D. dico pondus C à potentia |
|
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|
3811 in D moueri. fiat vt DA ad AB, ita pondus E ad potentiam in |
|
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|
3812 D; & $it pondus E ex puncto B $u$pen$um: potentia in D pondus |
|
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|
3813 E $u$tinebit. $ed DA ad AB maiorem habet proportionem, |
|
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|
3814 quàm C ad potentiam in D; & vt DA ad AB, ita e$t pondus E |
|
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|
3815 ad potentiam in D; pondus igitur E ad potentiam, quæ e$t in D, |
|
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|
3816 maiorem habebit proportionem, quàm pondus C ad eandem po |
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3817 tentiam. quare pondus E maius e$t pondere C. & cùm poten- |
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3818 tia in D pondus E $u$tineat, potentia igitur in D pondus C in B |
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3819 appen$um vecte AB, cuius fulcimentum e$t A, mouebit. quod |
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|
3820 demon$trare oportebat. |
|
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|
3821 <head>ALITER.</head> |
|
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|
3822 <p>Sit vectis AB, & |
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|
3823 pondus C in A ap- |
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|
3824 pen$um & poten- |
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|
3825 tia in B; $it qué fulci- |
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|
3826 mentum D: & DB |
|
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|
3827 <fig> |
|
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|
3828 ad DA maiorem habeat proportionem, quàm pondus C ad po |
|
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|
3829 tentiam in B. dico pondus C à potentia in B moueri. fiat BE ad |
|
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3830 EA, vt pondus C ad potentiam, erit punctum E inter BD. opor |
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3831 tet enim BE ad EA minorem habere proportionem, quàm DB |
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3832 ad DA, & ideo BE minor erit BD. & quoniam potentia in B $u <marg>1 <I>Huius.</I></marg> |
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3833 $tinet pondus C in A appen$um uecte AB, cuius fulcimentũ E; minor |
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3834 igitur potentia in B, quàm data, idem pondus $u$tinebit fulcimen |
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3835 to D. data ergo potentia in B pondus C mouebit uecte AB, cuius |
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3836 $ulcimentum e$t D. |
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|
3837 <foot>P</foot> |
|
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|
3838 <pb> |
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3839 <p>Sit deinde vectis AB, & fulci |
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|
3840 mentum A, & pondus C in D |
|
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|
3841 appen$um, $itq; potentia in B; & |
|
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3842 AB ad AD maiorem habeat pro- |
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|
3843 portionem, quàm pondus C ad |
|
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|
3844 potentiam in B. dico pondus C |
|
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|
3845 <fig> |
|
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|
3846 à potentia in B moueri. Fiat AB ad AE, vt pondus C ad poten |
|
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|
3847 <marg>8 <I>quinti.</I></marg> tiam; erit $imiliter punctum E inter BD. nece$$e e$t enim AE |
|
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|
3848 maiorem e$$e AD. & $i pondus C e$$et in E appen$um, potentia |
|
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|
3849 <marg>2 <I>Huius.</I></marg> in B illud $u$tineret. minor autem potentia in B, quàm data, $u$ti- |
|
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3850 <marg>1 <I>Cor.</I></marg> net pondus C in D appen$um; data ergo potentia in B pondus C in |
|
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3851 <marg>2 <I>Huius.</I></marg> D appen$um vecte AB, cuius $ulcimentum e$t A, mouebit. |
|
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|
3852 <p>Sit rur$us vectis AB, cu |
|
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|
3853 ius fulcimentum A, & pon |
|
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|
3854 dus C in B $it appen$um; |
|
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|
3855 $itq; potentia in D; & DA |
|
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|
3856 ad AB maiorem habeat |
|
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|
3857 <fig> |
|
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|
3858 proportionem, quàm pondus C ad potentiam in D. dico pon- |
|
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|
3859 dus C à potentia in D moueri. fiat vt pondus C ad potentiam, |
|
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|
3860 <marg>8 <I>Quinti.</I></marg> ita DA ad AE; erit AE maior AB; cùm maior $it proportio |
|
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|
3861 DA ad AB, quàm DA ad AE. & $i pondus C appendatur in |
|
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|
3862 <marg>3 <I>Huius.</I></marg> E, patet potentiam in D $u$tinere pondus C in E appen$um. mi- |
|
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|
3863 <marg>1 <I>Cor.</I></marg> nor autem potentia, quàm data, $u$tinet idem pondus C in B; |
|
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|
3864 <marg>3 <I>Huius.</I></marg> data igitur potentia in D pondus C in B appen$um mouebit ve- |
|
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|
3865 cte AB, cuius fulcimentum e$t A. quod oportebat demon- |
|
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|
3866 $trare. |
|
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|
3867 <head>PROPOSITIO XII.</head> |
|
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|
3868 <head>PROBLEMA.</head> |
|
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|
3869 <p>Datum pondus à data potentia dato vecte |
|
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|
3870 moueri. |
|
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|
3871 <pb n=58> |
|
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|
3872 <fig> |
|
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|
3873 <p>Sit pondus A vt centum, potentia verò mouens $it vt decem; |
|
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|
3874 $itq; datus vectis BC. oportet potentiam, quæ e$t decem pondus |
|
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|
3875 A centum vecte BC mouere. Diuidatur BC in D, ita vt CD |
|
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|
3876 ad DB eandem habeat proportionem, quàm habet centum ad |
|
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|
3877 decem, hoc e$t decem ad vnum; etenim $i D ficret fulcimentum, |
|
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|
3878 con$tat potentiam vt decem in C æqueponderare ponderi A in B <marg>1 <I>Huius.</I></marg> |
|
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|
3879 appen$o: hoc e$t pondus A $u$tinere. accipiatur inter BD quod |
|
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|
3880 uis punctum E, & fiat E fulcimentum. Quoniam enim maior <marg><I>Lemma huius.</I></marg> |
|
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|
3881 e$t proportio CE ad EB, quàm CD ad DB; maiorem habebit |
|
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|
3882 proportionem CE ad EB, quàm pondus A ad potentiam decem |
|
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|
3883 in C: potentia igitur decem in C pondus A centum in B appen- |
|
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|
3884 $um vecte BC, cuius fulcimentum $it E, mouebit. <marg>11 <I>Huius.</I></marg> |
|
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|
3885 <p>Si verò $it vectis |
|
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|
3886 BC, & fulcimen- |
|
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|
3887 tum B. diuidatur CB |
|
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|
3888 in D, ita vt CB ad |
|
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|
3889 BD eandem habeat |
|
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|
3890 proportionem, quã |
|
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|
3891 <fig> |
|
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|
3892 habet centum ad decem: & $i pondus A in D $u$pendatur, & po- |
|
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|
3893 tentia in C, potentia vt decem in C pondus A in D appen$um $u <marg>2 <I>Huius.</I></marg> |
|
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|
3894 $tinebit. accipiatur inter DB quoduis punctum E, ponaturq; pon |
|
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|
3895 dus A in E; & cùm $it maior proportio CB ad BE, quàm <marg>8 <I>Quinti.</I></marg> |
|
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|
3896 BC ad BD; maiorem habebit proportionem CB ad BE, quàm |
|
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|
3897 pondus A centum ad potentiam decem. potentia igitur decem <marg>11 <I>Huius.</I></marg> |
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3898 in C pondus A centum in E appen$um mouebit vecte BC, cu |
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3899 ius fulcimentum e$t B. quod facere oportebat. |
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3900 <foot>P 2</foot> |
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3901 <pb> |
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3902 <p>Hoc autem fieri non po- |
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3903 te$t exi$tente vecte BC, cuius |
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3904 fulcimentum $it B, & pondus |
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3905 A centum in C appen$um: po |
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3906 natur enim potentia $u$tinens |
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3907 pondus A vtcunq; inter BC, |
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3908 <marg>2 <I>Cor.</I></marg> vt in D, $emper potentia ma |
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3909 <marg>3 <I>Huius.</I></marg> ior erit pondere A. quare opor |
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3910 <fig> |
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3911 tet datam potentiam maiorem e$$e pondere A. $it igitur poten- |
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3912 tia data vt centum quinquaginta. diuidatur BC in D, ita vt CB |
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3913 ad BD $it, vt centum quinquaginta ad centum; hoc e$t tria ad duo: |
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3914 <marg>3 <I>Huius.</I></marg> & $i ponatur potentia in D, patet potentiam in D $u$tinere pon- |
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3915 dus A in C appep$um. accipiatur itaq; inter DC quoduis pun- |
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3916 <marg>8 <I>Quinti.</I></marg> ctum E, ponaturq; potentia mouens in E; & cùm maior $it pro- |
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3917 portio EB ad BC, quàm DB ad BC; habebit EB ad BC maio |
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3918 rem proportionem, quàm pondus A ad potentiam in E. poten |
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3919 <marg>11 <I>Huius.</I></marg> tia igitur vt centum quinquaginta in E pondus A centum in C |
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3920 appen$um vecte BC, cuius fulcimentum e$t B, mouebit. quod |
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3921 facere oportebat. |
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3922 <head>COROLLARIVM.</head> |
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3923 <p>Hinc manife$tum e$t $i data potentia $it dato |
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3924 pondere maior; hoc fieri po$$e, $iue ita exi$ten |
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3925 te vecte, vt eius fulcimentum $it inter pondus, |
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3926 & potentiam; $iue pondus inter fulcimentum, |
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3927 & potentiam habente; $iue demum potentia in- |
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|
3928 ter pondus, & fulcimentum con$tituta. |
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3929 <p>Sin autem data potentia minor, vel æqualis |
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3930 dato pondere fuerit; palam quoq; e$t id ip$um |
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3931 dumtaxat a$$e qui po$$e vecte ita exi$tente, vt eius |
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3932 fulcimentum $it inter pondus, & pontentiam; |
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|
3933 <pb n=59> |
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3934 vel pondus intra fulcimentum, & potentiam |
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|
3935 habente. |
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|
3936 <head>PROPOSITIO XIII.</head> |
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|
3937 <head>PROBLEMA.</head> |
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3938 <p>Quotcunq; datis in vecte ponderibus vbicun- |
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3939 què appen$is, cuius fulcimentum $it quoq; da- |
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3940 tum, potentiam inuenire, quæ in dato puncto |
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|
3941 data pondera $u$tineat. |
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|
3942 <fig> |
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3943 <p>Sint data pondera ABC in vecte DE, cuius fulcimentum F, |
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3944 vbicunq; in punctis DGH appen$a: collocandaq; $it potentia in |
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3945 puncto E. potentiam inuenire oportet, quæ in E data pondera |
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3946 ABC vecte DE $u$tineat. diuidatur DG in k, ita vt Dk ad KG |
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3947 $it, vt pondus B ad pondus A; deinde diuidatur kH in L, ita vt kL |
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3948 ad LH, $it vt pondus C ad pondera BA; atq; vt FE ad FL, ita |
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3949 fiant pondera ABC $imul ad potentiam, quæ ponatur in E. di- |
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|
3950 co potentiam in E data pondera ABC in DGH appen$a vecte |
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|
3951 DE, cuius fulcimentum e$t F, $u$tinere. Quoniam enim $i ponde |
|
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3952 ra ABC $imul e$$ent in L appen$a, potentia in E data pondera <marg>1 <I>Huius.</I></marg> |
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|
3953 in L appen$a $u$tineret; pondera verò ABC tàm in L ponderant, <marg>5 <I>Huius. de libra.</I></marg> |
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3954 quàm $i C in H, & BA $imul in K e$$ent appen$a; & AB in k tàm |
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|
3955 <pb> |
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|
3956 <fig> |
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3957 ponderant, quàm $i A in D, & B in G appen$a e$$ent; ergo po- |
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3958 tentia in E data pondera ABC in DGH appen$a vecte DE, cu- |
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3959 ius fulcimentum e$t F, $u$tinebit. Si autem potentia in quouis |
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|
3960 alio puncto vectis DE (præterquàm in F) con$tituenda e$$et, |
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3961 vt in k; fiat vt Fk ad FL, ita pondera ABC ad potentiam: $i- |
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3962 <marg>2 <I>Huius.</I></marg> militer demon$trabimus potentiam in k pondera ABC in pun- |
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3963 ctis DGH appen$a $u$tinere. quod facere oportebat. |
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|
3964 <fig> |
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3965 <p>Ex hac, & ex quinta huius, $i pondera ABC $int in vecte |
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3966 DE quomodocunq; po$ita; oporteatq; potentiam inuenire, quæ |
|
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3967 in E data pondera $u$tinere debeat: ducantur à centris grauita- |
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3968 tum ponderum ABC horizontibus perpendiculares, quæ ve- |
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|
3969 ctem DE in DGH punctis $ecent; cæteraq; eodem modo fiant: |
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3970 Manife$tum e$t, potentiam in E, vel in K data pondera $u$tinere. |
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3971 idem enim e$t, ac $i pondera in DGH e$$ent appen$a. |
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|
3972 <pb n=60> |
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|
3973 <head>PROPOSITIO XIIII.</head> |
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3974 <head>PROBLEMA.</head> |
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3975 <p>Data quotcunq; pondera in dato vecte vbi- |
|
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|
3976 cunq; & quomodocunq; po$ita à data potentia |
|
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|
3977 moueri. |
|
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|
3978 <fig> |
|
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3979 <p>Sit datus vectis DE, & $int data pondera vt in præcedentico |
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3980 rollario; $itq; A vt centum, B vt quinquaginta, C vt triginta; |
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3981 dataq; potentia $it vt triginta. exponantur eadem, inueniaturq; |
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3982 punctum L; deinde diuidatur LE in F, ita vt FE ad FL $it, vt |
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3983 centum octoginta ad triginta, hoc e$t $ex ad vnum: & $i F fieret |
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3984 fulcimentum, potentia vt triginta in E $u$tineret pondera ABC. <marg>13 <I>Huius.</I></marg> |
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3985 accipiatur igitur inter LF quoduis punctum M, fiatq; M fulci- |
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3986 mentum: manife$tum e$t potentiam in E vt triginta pondera <marg>11 <I>Huius.</I></marg> |
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3987 ABC vt centum octoginta vecte DE mouere. quod facere |
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3988 oportebat. |
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3989 <p>Hoc autem vniuersè a$$equi minimè poterimus, $i in extremita- |
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3990 te vectis fulcimentum e$$et, vt in D; quia proportio DE, ad DL |
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3991 hoc e$t proportio ponderum ABC ad potentiam, quæ pondera |
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3992 $u$tinere debeat, $emper e$t data. quod multo quoq; minus fieri |
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3993 po$$et, $i ponenda e$$et potentia inter DL. |
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3994 <pb> |
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3995 <head>PROPOSITIO XV.</head> |
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3996 <head>PROBLEMA.</head> |
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3997 <p>Quia verò dum pondera vecte mouentur, |
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3998 vectis quoq; grauitatem habet, cuius nulla ha- |
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3999 ctenus mentio facta e$t: idcirco primùm quo- |
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4000 modo inueniatur potentia, quæ in dato puncto |
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4001 datum vectem, cuius fulcimentum $it quoq; da- |
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4002 tum, $u$tineat, o$tendamus. |
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4003 <fig> |
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4004 <p>Sit datus vectis AB, cuius fulcimentum $it datum C; $itq; |
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4005 punctum D, in quo collocanda $it potentia, quæ vectem AB $u |
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4006 $tinere debeat, ita vt immobilis per$i$tat. ducatur à puncto C |
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4007 linea CE horizonti perpendicularis, quæ vectem AB in duas di- |
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4008 uidat partes AE EF, $itq; partis AE centrum grauitatis G, & |
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4009 partis EF centrum grauitatis H; à punctisqué GH horizon- |
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4010 tibus perpendiculares ducantur Gk HL, quæ lineam AF |
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4011 in punctis KL $ecent. quoniam enim vectis AB à linea CE in duas |
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4012 diuiditur partes AE EF; ideo vectis AB nihil aliud erit, ni$i |
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4013 duo pondera AE EF in vecte, $iue libra AF po$ita; cuius $u- |
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4014 $pen$io, $iue fulcimentum e$t C. quare pondera AE EF ita erunt |
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4015 po$ita, ac $i in kL e$$ent appen$a. diuidatur ergo kL in M, |
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4016 ita vt kM ad ML, $it vt grauitas partis EF ad grauitatem par- |
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4017 tis AE; & vt CA ad CM, ita fiat grauitas totius vectis AB ad |
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4018 potentiam, quæ $i collocetur in D (dummodo DA horizonti |
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4019 <pb n=61> |
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4020 perpendicularis exi$tat) vecti æqueponderabit; hoc e$t vectem <marg>13 <I>Huius.</I></marg> |
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4021 AB deor$um premendo $u$tinebit. quod inuenire oportebat. |
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4022 <p>Si verò potentia in puncto B ponenda e$$et. fiat vt CF ad CM |
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4023 ita pondus AB ad potentiam. $imili modo o$tendetur poten- |
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4024 tiam in B vectem AB $u$tinere. $imiliterq; demon$trabitur in quo- |
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4025 cunq; alio $itu (præterquàm in e) ponenda fuerit potentia, vt in |
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4026 N. fiat enim vt CO ad CM, ita AB ad potentiam; quæ $i pona- |
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4027 tur in N, vectem AB $u$tinebit. |
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4028 <p>Adiiciatur autem pondus in vecte appen$um, |
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4029 $iue po$itum; vt iisdem po$itis $it pondus P in |
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4030 A appen$um; potentiaq; $it ponenda in B, ita |
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4031 vt vectem AB vnà cum pondere P $u$tineat. |
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4032 <fig> |
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4033 <p>Diuidatur AM in Q, ita vt AQ ad QM $it, ut grauitas ue- |
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4034 ctis AB ad grauitatem ponderis P; deinde ut CF ad CQ, ita fat |
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4035 grauitas AB, & P $imul ad potentiam, quæ ponatur in B: patet |
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4036 potentiam in B uectem AB unà cum pondere P $u$tinere. Si ue- <marg>13 <I>Huius.</I></marg> |
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4037 rò e$$et CA ad CM, vt AB ad P; e$$et punctum C eorum centrum <marg><I>Ex $exta</I></marg> |
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4038 grauitatis, & ideo vectis AB vná cum pondere P ab$q; potentia in <marg>1 <I>Arch. de æquep.</I></marg> |
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4039 B manebit. $ed $i ponderum grauitatis centrum e$$et inter CF, vt |
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4040 in O; fiat vt CF ad CO, ita AB&P $imul ad potentiam, quæ |
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4041 in B, & vectem AB, & pondus P $u$tinebit. |
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4042 <foot>Q</foot> |
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4043 <pb> |
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4044 <fig> |
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4045 <p>Similiter o$tendetur, $i plura e$$ent pondera in vecte AB ubi- |
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4046 cunq;, & quomodocunq; po$ita. |
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4047 <p>In$uper ex his non $olum, ut in decimaquarta huius docuimus, |
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4048 quomodo $cilicet data pondera ubicunq; in uecte po$it a data poten |
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4049 tia dato uecte mouere po$$umus, eodem modo grauitate uectis |
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4050 con$iderata idem facere poterimus; uerùm etiam accidentia reli- |
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4051 qua, quæ $upra ab$q; uectis grauitatis con$ideratione demon$tra- |
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4052 <*>a $unt; $imili modo uectis grauitate con$iderata vná cum ponde |
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4053 ribus, uel $ine ponderibus o$tendentur. |
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4054 <pb n=62> |
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4055 <head>DE TROCHLEA.</head> |
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4056 <p>Trochleae in$trumento pon |
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4057 dus multipliciter moueri pote$t; |
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4058 quia verò in omnibus e$t eadem |
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4059 ratio: ideo (vt res euidentior ap- |
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4060 pareat) in iis, quæ dicenda $unt, |
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4061 intelligatur pondus $ur$um ad re |
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4062 ctos horizontis plano angulos hoc modo $em- |
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4063 per moueri. |
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4064 <foot>Q 2</foot> |
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4065 <pb> |
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4066 <p>Sit pondus A, quod ip$i ho |
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4067 rizontis plano $ur$um ad rectos |
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4068 angulos $it attollendum; & vt |
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4069 fieri $olet, trochlea duos habens |
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4070 orbiculos, quorum axiculi $int |
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4071 in BC, $upernè appendatur; |
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4072 trochlea verò duos $imiliter ha |
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4073 bens orbiculos, quorum axicu- |
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4074 li $int in DE, ponderi alligetur: |
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4075 ac per omnes vt riu$q; trochleæ |
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4076 orbiculos circunducatur ducta- |
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4077 rius funis, quem in altero eius ex |
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4078 tremo, putá in F, oportet e$$e |
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4079 religatum. potentia autem mo |
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4080 uens ponatur in G, quæ dum |
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4081 de$cendit, pondus A $ur$um ex |
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4082 aduer$o attolletur; quemadmo |
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4083 dum Pappus in octauo libro Ma |
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4084 thematicarum collectionum a$- |
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4085 $erit; nec non Vitruuius in deci |
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4086 mo de Architectura, & alii. |
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4087 <fig> |
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4088 <p>Quomodo autem hoc trochleæ in$trumen- |
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4089 tum reducatur ad vectem; cur magnum pondus |
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4090 ab exigua virtute, & quomodo, quantoq; in tem |
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4091 pore moueatur; cur funis in vno capite debeat |
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4092 e$$e religatus; quodq; $uperioris, inferioris&qacute;ue |
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4093 tro chleæ fuerit officium; & quomodo omnis in |
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4094 <pb n=63> |
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4095 numeris data proportio inter potentiam, & pon |
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4096 dus inueniri po$sit; dicamus. |
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4097 <head>LEMMA.</head> |
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4098 <p>Sint rectæ lineæ AB CD parallelæ, quæ in |
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4099 punctis AC circulum ACE contingant, cuius |
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4100 centrum F: & FA FC connectantur. Dico |
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4101 AFC rectam lineam e$$e. |
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4102 <p>Ducatur FE ip$is AB CD æquidi$tans. |
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4103 & quoniam AB, & FE $unt parallelæ, & |
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4104 angulus BAF e$t rectus; erit & AFE re- |
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4105 ctus. eodemq; modo CFE rectus erit. li- <marg>18 <I>Tertii.</I></marg> |
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4106 neaigitur AFC recta e$t. quod erat de- <marg>29 <I>Primi.</I></marg> |
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4107 mon$trandum. <marg>14 <I>Primi.</I></marg> |
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4108 <fig> |
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4109 <head>PROPOSITIO I.</head> |
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4110 <p>Si funis trochleæ $upernè appen$æ orbiculo |
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4111 circunducatur, alterumq; eius extremum pon- |
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4112 deri alligetur, altero interim à potentia pondus |
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4113 $u$tinente apprehen$o: erit potentia ponderi |
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4114 æqualis. |
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4115 <pb> |
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4116 <p>Sit pondus A, |
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4117 cui alligatus $it fu- |
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4118 nis in B; trochleaq; |
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4119 habens orbiculum C |
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4120 EF, cuius centrum |
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4121 D, $ur$um appenda- |
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4122 tur; $itq; D quoq; |
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4123 centrum axiculi; & |
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4124 circa orbiculum uo- |
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4125 luatur funis BC EF |
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4126 G; $itq; potentia |
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4127 in G $u$tinens pon- |
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4128 dus A. dico poten- |
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4129 tiam in G ponderi A |
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4130 æqualem e$$e. Sit FG |
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4131 æquidi$tans CB. |
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4132 Quoniam igitur pon |
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4133 <marg>1 <I>Huius. de libra.</I></marg> dus A manet; erit |
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4134 <fig> |
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4135 CB horizonti plano perpendicularis <*> quare FG eidem plano per- |
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4136 <marg>8 <I>Vndecimi.</I></marg> pendicularis erit. Sint CF pũcta in orbiculo, à quibus funes CB FG |
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4137 in horizontis planũ ad rectos angulos de$cendunt; tangent BC FG |
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4138 orbiculũ CEF in punctis CF. orbiculũ enim $ecarenõ po$$unt. con |
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4139 nectantur DC DF; erit CF recta linea, & anguli DCB DFG recti. |
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4140 <marg>18 <I>Tertii.</I></marg> Quoniã aut&etilde; BC tùm horizonti, tùm ip$i CF e$t perpendicularis; |
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4141 erit linea CF horizonti æquidi$tans. cùm verò põdus appen$um $it |
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4142 <marg><I>Ex</I> 28 <I>Pri mi.</I></marg> in BC, & potentia $it in G; quod idem e$t, ac $i e$$et in F; erit |
|
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4143 CF tanquam libra, $iue vectis, cuius centrum, $iue fulcimentum e$t |
|
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4144 D; nam in axiculo orbuculus $u$tinetur; atq; punctum D, cùm $it |
|
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4145 centrum axiculi, & orbiculi, etiam vtri$que circumuolutis |
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4146 immobile remanet. Itaq; cùm di$tantia DC $it æqualis di$tantiæ |
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4147 DF, potentiaq; in F ponderi A in C appen$o æqueponderet, cùm |
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4148 <marg>1 <I>Primi. Archim. de æquepond.</I></marg> pondus $u$tineat, ne deor$um vergat; erit potentia in F, $iue in G |
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4149 (nam idem e$t) con$tituta ponderi A æqualis. Idem enim effi- |
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4150 cit potentia in G, ac $i in G aliud e$$et appen$um pondus æquale |
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4151 ponderi A; quæ pondera in CF appen$a æquæponderabunt. Præ- |
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4152 terea, cùm in neutram fiat motus partem, idem erit vnico exi- |
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4153 <pb n=64> |
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4154 $tente fune BC EFG hoc modo orbiculo circumuoluto, ac $i duo |
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4155 e$$ent funes BC FG alligati in vecte, $iue libra CF. |
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4156 <head>COROLLARIVM.</head> |
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4157 <p>Ex hoc manife$tum e$$e pote$t, idem pon- |
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4158 dus ab eadem potentia ab$q; ullo huius tro- |
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4159 chleæ auxilio nihilominus $u$tineri po$$e. |
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4160 <p>Sit enim pondus H æquale |
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4161 ponderi A, cui alligatus $it funis |
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4162 kL; $itq; potentia in L $u$tinens |
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4163 pondus H. cùm autem pondus |
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4164 ab$q; vllo adminiculo $u$tinere |
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4165 volentes tanta vi opus $it, quanta |
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4166 ponderi e$t æqualis; erit potentia |
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4167 in L ponderi H æqualis; pondus |
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4168 verò H ip$i ponderi A e$t æquale, |
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4169 cui potentia in G e$t æqualis; erit |
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4170 igitur potentia in G potentiæ in L |
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4171 æqualis. quod idem e$t, ac $i ead&etilde; |
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4172 potentia idem pondus $u$tineret. |
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4173 <fig> |
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4174 <p>Præterea $i potentiæ in G, & |
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4175 in L inuicem fuerint æquales, $eor |
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4176 $um autem ponderibus minores; |
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4177 patet potentias ponderibus $u$ti- |
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4178 nendis non $ufficere. $i verò maiores, manife$tum e$t pondera à |
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4179 pontentiis moueri. & $ic in eadem e$$e proportione potentiam in |
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4180 L. ad pondus H, veluti potentia in G ad pondus A. |
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4181 <p>Sed quoniam in demon$tratione a$$umptum fuit axiculum cir- |
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4182 cumuerti, qui vt plurimum immobilis manet; idcirco immobili |
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4183 quoq; manente axiculo idem o$tendatur. |
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4184 <pb> |
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4185 <p>Sit orbiculus trochleæ CEF, cu |
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4186 ius centrum D; $itq; axiculus GHk, |
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4187 cuius idem $it centrum D. Ducatur |
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4188 CG DkF diameter horizonti æ- |
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4189 quidi$tans. & quoniam dum orbi- |
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4190 culus circumuertitur, circumferen- |
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4191 tia circuli CEF $emper e$t æquidi- |
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4192 $tans circumferentiæ axiculi GHk; |
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4193 circa enim axiculum circumuerti- |
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4194 tur; & circulorum æquidi$tantes cir |
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4195 cumferentiæ idem habent centrum; |
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4196 erit punctum D $emper & orbiculi, |
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4197 <fig> |
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4198 & axiculi centrum. Itaq; cùm DC $it æqualis DF, & DG ip$i |
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4199 Dk; erit GC ip$i kF æqualis. $i igitur in vecte, $iue libra CF |
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4200 pondera appendantur æqualia, æqueponderabunt. di$tantia enim |
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4201 CG æqualis e$t di$tantiæ kF; axiculu$<*>; GHK immobilis gerit |
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4202 vicem centri, $iue fulcimenti. immobili igitur manente axicu- |
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4203 lo, $i ponatur in F potentia $u$tinens pondus in C appen$um; erit |
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4204 potentia in F ip$i ponderi æqualis. quod erat o$tendendum. |
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4205 <p>Et cùm idem pror$us $it, $iue axiculus circumuertatur, $iue mi- |
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4206 nus; liceat propterea in iis, quæ dicenda $unt, loco axiculi cen- |
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4207 trum tantùm accipere. |
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4208 <head>PROPOSITIO II.</head> |
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4209 <p>Si funis orbiculo trochleæ ponderi alligatæ |
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4210 circumducatur, altero eius extremo alicubi reli- |
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4211 gato, altero uerò à potentia pondus $u$tinente |
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4212 apprehen$o; erit potentia ponderis $ubdupla. |
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4213 <pb n=65> |
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4214 <p>Si pondus A; $it BCD |
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4215 orbiculus trochleæ pon- |
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4216 deri A alligate, cuius cen |
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4217 trum E; funis deinde FB |
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4218 CDG circa orbiculum |
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4219 voluatur, qui religetur in |
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4220 F; $itq; potentia in G $u |
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4221 $tinens pondus A. dico |
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4222 potentiam in G $ubdu- |
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4223 plam e$$e ponderis A. $int |
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4224 funes FB GD puncti E |
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4225 horizonti perpendicula- |
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4226 res, qui inter $e $e æqui- <marg>6 <I>Vndecimi</I></marg> |
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4227 di$tantes erunt; tangantq; |
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4228 funes FB GD circulum |
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4229 BCD in BD punctis. |
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4230 connectatur BD; erit BD |
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4231 per centrum E ducta, <marg><I>Ex præcedenti.</I></marg> |
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4232 <fig> |
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4233 ip$iu$qué centri horizonti æquidi$tans. Cùm autem potén- |
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4234 tia in G trochlea pondus A $u$tinere debeat, funem ex altero ex- |
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4235 tremo religatum e$$e oportet, puta in F; ita vt F æqualiter $altem |
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4236 potentiæ in G re$i$tat, alioquin potentia in G nullatenus pondus |
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4237 $u$tinere po$$et. Et quoniam potentia fune $u$tinet orbiculum, |
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4238 qui reliquam trochleæ partem, cui appen$um e$t pondus, $u$tinet |
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4239 axiculo; grauitabit hæc trochleæ pars in axiculo, hoc e$t in centro |
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4240 E. quare pondus A in eodem quoq; centro E ponder<*>bit, ac $i |
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4241 in E e$$et appen$um. po$ita igitur potentia, quæ in G, vbi D |
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4242 (idem enim pror$us e$t) erit BD tanquam vectis, cuius fulci |
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4243 mentum erit B, pondus in E appen$um, & potentia in D. con |
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4244 uenienter enim fulcimenti rationem ip$um B $ubire pote$t, exi |
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4245 $tente fune FB immobili. cæterum hoc po$terius magis eluce$cet. |
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4246 Quoniam autem potentia ad pondus eandem habet proportio- <marg>2 <I>Huius de vecte.</I></marg> |
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4247 nem, quàm BE ad BD; & BE in $ubdupla e$t proportione |
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4248 ad BD: potentia igitur in G ponderis A $ubdupla erit. quod de- |
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4249 mon$trare oportebat. |
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4250 <foot>R</foot> |
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4251 <pb> |
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4252 <p>Hoc igitur ita $e ha- |
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4253 bet vnico exi$tent efune |
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4254 FBC DG ip$i orbi culo |
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4255 circumducto, ac $i duo e$ |
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4256 $ent funes BF GD ve- |
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4257 cti BD alligati, cuius ful |
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4258 cimentum erit B, pon- |
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4259 dus in E appen$um, & |
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4260 potentia $u$tinens in D, |
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4261 vel quod idem e$t in G. |
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4262 <fig> |
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4263 <head>COROLLARIVM I.</head> |
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4264 <p>Ex hoc itaq; manife$tum e$t, pondus hoc mo |
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4265 do à minori in $ubdupla proportione potentia |
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4266 $u$tineri, quam $ine vllo huiu$inodi trochleæ |
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4267 auxilio. |
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|
4268 <pb n=66> |
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4269 <p>Veluti $it pondus H ponderi A |
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4270 æquale, cui religatus $it funis kL; |
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4271 potentiaq; in L $u$tineat pondus H; |
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4272 erit potentia in L $eor$um ponderi |
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4273 H, & ponderi A æqualis; $ed poten |
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4274 tia in G $ubdupla e$t ponderis A, |
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4275 quare potentia in G $ubdupla erit po |
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4276 tentiæ, quæ e$t in L. & ho cmodo in |
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4277 huiu$cemodi reliquis omnibus pro |
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4278 portio inueniri poterit. |
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|
4279 <fig> |
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|
4280 <head>COROLLARIVM. II.</head> |
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4281 <p>Manife$tum e$t etiam; $i duæ fuerint poten- |
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4282 tiæ vna in G, altera in F, pondus A $u$tinentes; |
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4283 vtra$q; $imul ponderi A æquales e$$e: & vnam |
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4284 quamque $u$tinere dimidium ponderis A. |
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4285 <p>Hoc autem ex tertio, & quarto corollario $ecundæ huius in |
|
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|
4286 tractatu de vecte patet. |
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|
4287 <head>COROLLARIVM III.</head> |
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4288 <p>Illud quoq; præterea innote$cit, cur $cilicet fu |
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4289 nis ex altero religatus e$$e debeat extremo. |
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|
4290 <foot>R 2</foot> |
|
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4291 <pb> |
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4292 <head>PROPOSITIO III.</head> |
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4293 <p>Si vtri$q; duarum trochlearum $ingulis or- |
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4294 biculis, quarum altera $upernè, altera verò in- |
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4295 fernè con$tituta, ponderiq; alligata fuerit, cir |
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4296 cunducatur funis; altero eius extremo alicubi |
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4297 religato, altero verò à potentia pondus $u$ti- |
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4298 nente detento; erit potentia ponderis $ub du- |
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4299 pla. |
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4300 <p>Sit pondus A; $it BCD orbiculus trochleæ pon |
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4301 deri A alligatæ, cuius centrum K; EFG verò |
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4302 $it trochleæ $ur$um appen$æ, cuius centrum H. |
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4303 deinde LBC DME FGN funis circa orbicu- |
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4304 los ducatur, qui religetur in L; $itq; potentia in |
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4305 N $u$tinens pondus A. dico potentiam in N |
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4306 $ubduplam e$$e ponderis A. $i enim potentia $u |
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4307 $tinens pondus A vbi M collocata foret, e$$et |
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4308 vtiq; potentia in M $ubdupla ponderis A. po- |
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4309 <marg>2 <I>Huius.</I></marg> tentiæ verò in M æqualis e$t vis in N. e$t e- |
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4310 <marg>1 <I>Huius.</I></marg> nim ac $i potentia in M dimidium ponderis |
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4311 A $ine trochlea $u$tineret, cui æqueponderat |
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4312 pondus in N ponderis A dimidio æquale. |
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4313 quare vis in N æqualis dimidio ponderis A |
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4314 ip$um A $u$tinebit. Potentia igitur in N $u$ti |
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4315 nens pondus A $ubdupla e$t ip$ius A. quod |
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4316 demon$trare oportebat. |
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4317 <fig> |
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4318 <pb n=67> |
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4319 <p>Si verò vt in $ecunda figura $it fu |
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4320 nis BC DEF GHkL orbiculis cir |
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4321 cumuolutus, & religatus in B; poten |
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4322 tiaq; in L pondus A $u$tineat: erit |
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4323 potentia in L $imiliter ponderis $ubdu |
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4324 pla. orbiculus enim trochleæ $upe- |
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4325 rioris, ip$aqué trochlea penitus $unt |
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4326 inutiles: & idem e$t, ac $i funis reli |
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4327 gatus e$$et in F, & potentia in L $u |
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4328 $tineret pondus $ola trochlea ponderi |
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4329 alligata, quæ potentia ponderis A o$ten |
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4330 $a e$t $ubdupla. |
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4331 <fig> |
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4332 <head>COROLLARIVM.</head> |
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4333 <p>Ex his $equitur, $i duæ $int potentiæ in BL; |
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4334 vtra$q; inter $e $e æquales e$$e. |
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4335 <p>Vtraq; enim $eor$um e$t ip$ius A $ubdupla. |
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4336 <pb> |
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4337 <head>PROPOSITIO IIII.</head> |
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4338 <p>Sit vectis AB, cuius fulcimentum $it A; qui |
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4339 bifariam diuidatur in D: $itq; pondus C in D |
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4340 appen$um; duæq; $int potentiæ æquales in BD |
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4341 pondus C $u$tinentes. Dico unamquamq; poten |
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4342 tiam in BD ponderis C $ubtriplam e$$e. |
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4343 <p>Quoniam enim altera |
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4344 potentia e$t in D colloca |
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4345 ta, & pondus C in eodem |
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4346 puncto D e$t appen$um; |
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4347 potentia in D partem |
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4348 ponderis C $u$t^{i}nebit ip- |
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4349 $i potentiæ D æqualem. |
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4350 <fig> |
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4351 quare potentia in B partem $u$tinebit reliquam, quæ pars dupla erit |
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4352 ip$ius potentiæ in B; cùm pondus ad potentiam eandem habeat |
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4353 proportionem, quam AB ad AD: & potentiæ in BD $unt æqua- |
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4354 les; ergo potentia in B duplam $u$tinebit partem eius, quam $u$ti |
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4355 net potentia in D. diuidatur ergo pondus C in duas partes, qua |
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4356 rum vna $it reliquæ dupla; quod fiet, $i in tres partes æquales EFG |
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4357 diui$erimus: tunc enim FG dupla erit ip$ius E. Itaq; potentia |
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4358 in D partem E $u$tinebit, & potentiam in B reliquas FG. vtreq; |
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4359 igitur inter $e $e æquales potentiæ in BD $imul totum $u$tinebunt |
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4360 pondus C. & quoniam potentia in D partem E $u$tinet, quæ ter |
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4361 tia e$t pars ponderis C, ip$iq; e$t æqualis; erit potentia in D $ub |
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4362 tripla ponderis C. & cùm potentia in B $u$tineat partes FG, qua |
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4363 rum potentia in B e$t $ubdupla; erit in B potentia vni partium FG, |
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4364 putà G æqualis. G verò tertia e$t pars ponderis C; potentia |
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4365 igitur in B $ubtripla erit ponderis C. Vnaquæq; ergo potentia in |
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4366 BD $ubtripla e$t ponderis C. quod demon$trare oportebat. |
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4367 <pb n=68> |
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4368 <fig> |
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4369 <p>Et $i duo e$$ent vectes AB EF bifariam in GD diui$i, quorum |
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4370 fulcimenta e$$ent AF, & pondus C in DG vtriq; vecti appen- |
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4371 $um, ita tamen vt in vtroq; æqualiter ponderet; duæq; e$$ent |
|
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4372 æquales potentiæ in BG: eadem pror$us ratione o$tendetur, |
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4373 vnamquamq; potentiam in B, & G ponderis C $ubtriplam |
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4374 e$$e. |
|
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4375 <head>PROPOSITIO V.</head> |
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4376 <p>Si vtri$q; duarum trochlearũ $ingulis orbiculis, |
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4377 quarum altera $upernè, altera verò infernè con$ti |
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4378 tuta, ponderiq; alligata fuerit, circumducatur fu |
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4379 nis; altero eius extremo inferiori trochleæ reli- |
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4380 gato, altero verò à potentia pondus $u$tinente |
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4381 detento: erit potentia ponderis $ubtripla. |
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4382 <pb> |
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4383 <p>Sit pondus A; $it BCD orbiculus tro- |
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4384 chleæ ponderi A alligate, cuius centrum |
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4385 E; & FGH trochleæ $ur$um appen$æ, cu- |
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4386 ius centrum k; & LFGHBCDM funis |
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4387 orbiculis circumducatur, qui religetur in L |
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4388 trochleæ inferiori; $itq; potentia in M $u- |
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4389 $tinens pondus A. dico potentiam in M |
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4390 $ubtriplam e$$e ponderis A. ducantur FH |
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4391 BD per centra kE horizonti æquidi$tan- |
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4392 tes, $icut in præcedentibus dictum e$t Quo- |
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4393 niam enim funis FL trochleam $u$tinet in- |
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4394 feriorem, quæ $u$tinet orbiculum in eius |
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4395 centro E; erit funis in L vt potentia $u$ti- |
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4396 nens orbiculum, ac $i in ip$o E centro e$$et; |
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4397 potentia verò in M e$t, ac $i e$$et in D; |
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4398 efficietur igitur DB tanquam vectis, cuius |
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4399 <marg><I>In</I> 2 <I>Huius</I></marg> fulcimentum erit B; pondus verò A (vt $u |
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4400 pra o$ten$um e$t) ex E $u$pen$um à dua- |
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4401 bus potentiis altera in D, altera in E $u$ten |
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4402 tatum. Cùm autem in pondere $u$tinendo |
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4403 vectes FH BD immobiles maneant, $i in |
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4404 funibus FL HB appendantur pondera, e- |
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4405 <marg>1 <I>Huius.</I></marg> runt hæc ip$a æqualia; cùm vectis FH ha- |
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4406 beat fulcimentum in medio; alioquin ex al |
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4407 tera parte deor$um fieret motus, quod tam&etilde; |
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4408 non contingit. tam igitur $u$tinet funis FL, |
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4409 quàm HB. deinde quoniam ex medio ve- |
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4410 <fig> |
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4411 cte BD pondus $u$penditur, idcirco $i duæ fuerint potentiæ in BD |
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4412 <marg><I>Ex</I> 3 <I>Cor.</I> 2 <I>Huius ve cte.</I></marg> pondus $u$tinentes, erunt inuicem æquales. & quamquam funis |
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4413 <pb n=69> |
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4414 FL ip$e quoq; pondus $u$tineat, cùm potentiæ in E vic&etilde; gerat; quia |
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4415 tamen ex eodemmet puncto $u$tinet, vbi appen$um e$t pondus, non |
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4416 efficiet propterea, quin pot entiæ in BD $int inter $e $e æquales; |
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4417 opitulatur enim tàm vni, quàm alteri. potentiæ verò in BD eæ- |
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4418 dem $unt, ac $i e$$ent in HM; quare tàm $u$tinebit funis MD, |
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4419 quàm HB. ita verò $u$tinet HB, atq; FL; funis igitur MD ita |
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4420 $u$tinebit, $icut FL, hoc e$t, ac $i in D, & L appen$a e$$ent pon- |
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4421 dera æqualia. Cùm itaq; æqualia pondera à potentiis $u$tinean- |
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4422 tur æqualibus, potentiæ in ML æquales erunt; quarum eadem pror |
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4423 $us e$t ratio, ac $i e$$ent ambæ in DE. Itaq; cùm pondus A in |
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4424 medio vectis BD $it appen$um, duæq; potentiæ $int æquales in |
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4425 DE pondus $u$tinentes; erit B fulcimentum, ac vn aquæq; potentia, <marg>4 <I>Huius.</I></marg> |
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4426 $iue in DE, $iue in ML $ubtripla ponderis A. ergo potentia in M |
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4427 $u$tinens pondus $ubtripla erit ponderis A. quod o$tendere o- |
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4428 portebat. |
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4429 <head>COROLLARIVM.</head> |
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4430 <p>Ex hoc manife$tum e$t, vnumquemq; funem |
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4431 MD FL HB tertiam $u$tinere partem pon- |
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4432 deris A. |
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|
4433 <foot>S</foot> |
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|
4434 <pb> |
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4435 <p>Præterea, $ifunis ex M per a- |
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4436 lium adhuc deferatur orbiculum $u |
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4437 periorem in trochlea $ur$um $imi- |
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4438 liter appen$a con$titutum, cuius |
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4439 centrum N; ita vt perueniat in O; |
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4440 ibiq; à potentia detineatur; erit po |
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4441 tentia in O $u$tinens pondus A iti |
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4442 dem $ubtripla ip$ius ponderis. fu |
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4443 nis enim MD tantùm ponderis$u |
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4444 $tinet, ac $i in D appen$um e$$et |
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4445 pondus æquale tertiæ parti ponde |
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4446 <marg>1 <I>Huius.</I></marg> ris A, cui æquiualet potentia in |
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4447 O ip$i æqualis, hoc e$t $ubtripla |
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4448 ponderis A. Potentia igitur in O |
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4449 $ubtripla e$t ponderis A. |
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|
4450 <fig> |
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4451 <p>Et ne idem $æpius repetatur, no |
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4452 ui$$e oportet potentiam in O $em |
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4453 per æqualem e$$e ei, quæ e$t in M; |
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4454 hoc e$t $i potentia in M e$$et $ub |
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|
4455 quadrupla, $ubquintupla, vel huiu$ |
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|
4456 modi aliter ip$ius ponderis; poten |
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4457 tia quoq; in O erit itidem $ubqua |
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4458 drupla, $ubquintupla, atq; ita dein |
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|
4459 ceps eiu$demmet ponderis, quem |
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|
4460 madmodum $e habet potentia |
|
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|
4461 in M. |
|
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|
4462 <pb n=70> |
|
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|
4463 <head>PROPOSITIO VI.</head> |
|
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4464 <p>Sint duo vectes AB CD bifariam diui$i in |
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4465 EF, quorum fulcimenta $int. in BD; $itq; pon |
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|
4466 dus G in EF vtriq; vecti appen$um, ita ut ex |
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4467 vtroq; æqualiter ponderet; duæq; $int potentiæ |
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4468 in AC æquales pondus $u$tinentes. Dico unam |
|
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|
4469 quamq; potentiam in AC $ubquadruplam e$- |
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|
4470 $e ponderis G. |
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4471 <p>Cùm enim potentiæ in |
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|
4472 AC totum $u$tineant pon- |
|
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|
4473 dus G, potentiaq; in A ad |
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|
4474 partem ponderis, quod $u$ti |
|
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|
4475 net, $it vt BE ad BA; po- <marg>2 <I>Huius. de vecte.</I></marg> |
|
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|
4476 tentia verò in C ad partem |
|
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|
4477 ip$ius G, quod $u$tinet, ita |
|
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|
4478 $it vt DF ad DC; & vt BE |
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4479 ad BA, ita e$t DF ad DC; |
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4480 <fig> |
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4481 erit potentia in A ad partem ponderis, quod $u$tinet, vt poten- |
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4482 tia in C ad ip$ius ponderis, quod $u$tinet, partem; & potentiæ |
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4483 in AC $unt æquales; æquales igitur erunt partes ponderis G, |
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4484 quæ à potentiis $u$tinentur. quare vnaquæq; potentia in A C di- |
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4485 midium $u$tinebit ponderis G. Potentia verò in A $ubdupla e$t pon |
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4486 deris, quod $u$tinet: ergo potentia in A dimidio dimidii, hoc |
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4487 e$t quartæ portioni ponderis G æqualis erit; ideoq; $ubquadrupla |
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4488 erit ponderis G. neq; aliter demon$trabitur potentiam in C $ub-quadruplam |
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4489 e$$e eiu$dem ponderis G. quod demon$trare opor- |
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4490 tebat. |
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4491 <foot>S 2</foot> |
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4492 <pb> |
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4493 <p>Si verò tres $int vectes |
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4494 AB CD EF bifariam di- |
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4495 ui$i in GHk, quorum fulci |
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4496 menta $int BDF; & pondus |
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4497 L eodem modo in GHK |
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4498 appen$um; $intq; tres poten |
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4499 tiæ in ACE æquales pondus |
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4500 $u$tinentes; $imiliter o$ten |
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4501 detur vnamquamque po- |
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4502 tentiam $ub$excuplam e$$e |
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4503 ponderis L. atq; hoc ordi |
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4504 ne $i quatuor e$$ent vectes, |
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4505 & quatuor potentiæ; erit vnaquæq; potentia $uboctupla ponderis. |
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4506 atq; ita deinceps in infinitum. |
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4507 <fig> |
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4508 <head>PROPOSITIO VII.</head> |
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4509 <p>Si tribus duarum trochlearum orbiculis, quarũ |
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4510 altera $upernè vnico duntaxat, altera verò infer- |
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4511 nè duobus autem in$ignita orbiculis, ponderiq; |
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4512 alligata con$tituta fuerit, funis circumponatur; al |
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4513 tero eius extremo alicubi religato, altero verò à |
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4514 potentia pondus $u$tinente retento; erit potentia |
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4515 ponderis $ubquadrupla. |
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4516 <pb n=71> |
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4517 <p>Sit pondus A; $int tres orbiculi, quorum |
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4518 centra BCD; orbiculu$q;, cuius centrum D, |
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4519 $it trochleæ $ur$um appen$æ; quorum verò |
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4520 $unt centra BC, $int trochleæ ponderi A alli |
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4521 gatæ; funi$q; EFGHkLNOP per omnes |
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4522 circumducatur orbiculos, qui religetur in E; |
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4523 $itq; vis in P $u$tinens pondus A. dico po |
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4524 tentiam in P $ubquadruplam e$$e ponderis |
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4525 A. ducantur kL GF ON per rotularum |
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4526 centra, & horizonti æquidi$tantes, quæ (ex |
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4527 iis, quæ dicta $unt) tanquam vectes erunt. |
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4528 & quoniam propter vectem, $iue libram kL, |
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4529 cuius fulcimentum, $iue centrum e$t in me |
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4530 dio, tàm $u$tinet funis kG, quàm LN, cùm <marg>1 <I>Huius.</I></marg> |
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4531 in neutram partem fiat motus. nec non |
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4532 propter vectem GF, è cuius medio veluti $u |
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4533 $pen$um dependet onus; $i duæ e$$ent in GF |
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4534 potentiæ, $eu in HE (e$t enim par vtriu$q; |
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4535 $itus ratio, vt iam $epius dictum e$t) e$$ent <marg><I>Ex</I> 2 <I>Cor.</I> 2 <I>Huius.</I></marg> |
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4536 vtiq; huiu$modi potentiæ inuicem æquales. |
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4537 quare ita $u$tinet funis HG, vt EF. $imiliter |
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4538 o$ten detur funem PO tàm $u$tinere, quàm |
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4539 LN: quare funes PO kG EF LN æqua |
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4540 liter $u$tinent. æqualiter igitur funis PO $u |
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4541 $tinet, vt kG. $i ergo duæ intelligantur e$ |
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4542 <fig> |
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4543 $e potentiæ in OG, $eu in PH, quod idem e$t, pondus nihilomi |
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4544 nus $u$tinentes, quemadmodum funes $u$tinent, æquales vtiq; e$ |
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4545 $ent; & GF ON duorum vectium vires gerent; quorum fulci |
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4546 menta erunt FN, & pondus A in BC medio vectium appen$um. |
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4547 & quoniam omnes funes æqualiter $u$tinent, tàm $u$tinebunt |
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4548 duo PO LN, quàm duo KGEF; tàm igitur $u$tinebit vectis |
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4549 ON, quàm vectis GF. quare in vtroq; vecte ON GF æquali |
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4550 ter pondus põderabit. erit ergo vnaquæq; potentia in PH $ubquadru <marg>6 <I>Huius.</I></marg> |
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4551 pla ponderis A. & cùm funis KG potentiæ loco $umatur, quippè |
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4552 qui haud $ecus $u$tinet, quàm PO; erit potentia in P $u$tinens pon- |
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4553 dus A ip$ius ponderis $ubquadrupla. quod demon$trare oportebat. |
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4554 <pb> |
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4555 <head>COROLLARIVM I.</head> |
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4556 <p>Hinc manife$tum e$t vnumquemq; funem EF |
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4557 GK LN OP quartam $u$tinere partem pon- |
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4558 deris A. |
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4559 <head>COROLLARIVM II.</head> |
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4560 <p>Patet etiam orbiculum, cuius centrum C, |
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4561 non minus eo, cuius centrum e$t B, $u$tinere. |
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4562 <head>ALITER.</head> |
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4563 <p>Adhuc ii$dem po$itis, $i duæ e$$ent poten |
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4564 tiæ æquales pondus A $u$tinentes, vna in O |
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4565 <marg><I>Ex</I> 4 <I>Huius</I></marg> altera in C; e$$et vnaquæq; dictarum poten |
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4566 riarum ponderis A $ubtripla. $ed quoniam |
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4567 vectis GF, cuius fulcimentum e$t F bifariam |
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4568 diui$us e$t in C; $i igitur ponatur in G poten |
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4569 tia idem pondus $u$tinens, vt potentia in C; |
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4570 erit potentia in G $ubdupla potentiæ, quæ e$ |
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4571 $et in C; nam $i potentia in C $e ip$a pon- |
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4572 dus in C appen$um $u$tineret, e$$et vtiq; ip |
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4573 $i ponderi æqualis; & idem pondus, $i à po |
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4574 <marg>2 <I>Huius. de vecte.</I></marg> tentia in G $u$tineretur, e$$et ip$ius poten |
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4575 tiæ in G duplum; potentia veró in C $ubtri |
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4576 pla e$$et ponderis A; ergo potentia in G |
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4577 $ub$excupla e$$et ponderis A. Cùm itaq; |
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4578 potentia in O $ubtripla $it ponderis A, & |
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4579 potentia in G $ub$excupla; erunt vtræq; $i- |
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4580 mul potentiæ in OG ip$ius ponderis A $ub |
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4581 duplæ. tertia enim pars cum $exta dimi- |
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4582 dium efficit. quoniam autem potentiæ in |
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4583 OG, $iue in PH (vt prius dictum e$t) |
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4584 $unt inter $e æquales, ac vtræq; $imul $ubdu |
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4585 plæ $unt ponderis A. erit vnaquæq; poten |
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4586 <fig> |
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4587 <pb n=72> |
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4588 tia in P Hip$ius A $ubquadrupla. Potentia igitur in P $u$tinens pon |
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4589 dus A ip$ius ponderis A $ubquadrupla erit. quod erat o$ten- |
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4590 dendum. |
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4591 <p>Si verò funis religetur in E, |
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4592 & $ecundùm quatuor adhuc |
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4593 circumuoluatur orbiculos, per |
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4594 ueniatq; ad P. $imiliter o$ten |
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4595 detur potentiam in P $ubqua- |
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4596 druplam e$$e ponderis A. |
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4597 idem enim e$t, ac $i funis re- |
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4598 ligatus e$$et in L, potentiaq; |
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4599 $u$tineret pondus fune tribus |
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4600 tantùm orbiculis circumdu- |
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4601 cto, quorum centra e$$ent B |
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4602 CQ. orbiculus enim cuius |
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4603 centrum D e$t pœnitus inu- |
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4604 tilis. |
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4605 <fig> |
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4606 <pb> |
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4607 <head>PROPOSITIO VIII.</head> |
|
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4608 <p>Sint duo vetes AB CD bifariam diui$i in EF, |
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4609 quorum fulcimenta $int AC, & pondus G in |
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4610 punctis EF vtriq; vecti $it appen$um, ita vt ex |
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4611 vtroq; æqualiter ponderet; tre$q; $int potentiæ |
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4612 æquales in BDE pondus G $u$tinentes. Dico |
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4613 vnamquamq; $eor$um ex dictis potentiis $ub- |
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4614 quintuplam e$$e ponderis G. |
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4615 <p>Quoniam enim pondus G |
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4616 appen$um e$t in EF, & tres |
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4617 $unt potentiæ in EBD æqua |
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4618 les; ideo potentia in E partem |
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4619 tantùm ponderis G $u$tinebit |
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4620 ip$i potentiæ in E æqualem; |
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4621 potentiæ verò in BD partem |
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4622 $u$tinebunt reliquam; & pars, |
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4623 <marg>2 <I>Huius. de vecte.</I></marg> quam $u$tinet B, erit ip$ius |
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4624 dupla; pars autem, quam $u |
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4625 <fig> |
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4626 $tinet D, erit $imiliter ip$ius D dupla; propter proportionem |
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4627 BA ad AE, & DC ad CF. Cùm itaq; potentiæ in BD $int æqua |
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4628 <marg><I>In</I> 6 <I>Huius</I></marg> les, erunt (ex iis, quæ $upra dictum e$t) partes ponderis G, quæ |
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4629 à potentiis BD $u$tinentur, inter $e $e æquales; & vnaquæq; du |
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4630 pla eius partis, quæ à potentia in E $u$tinetur. diuidatur er- |
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4631 go pondus G in tres partes, quarum duæ $int inter $e $e æquales, |
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4632 nec non vnaquæq; $eor$um alterius tertiæ partis dupla. quod |
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4633 fiet, $i in quinq; partes æquales HKLMN diuidatur; pars |
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4634 enim compo$ita ex duabus partibus kL dupla e$t partis H; pars |
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4635 quoq; MN eiu$dem partis H e$t $imiliter dupla. quare & pars |
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4636 kL parti MN erit æqualis. Su$tineat autem potentia in E par |
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4637 tem H; & potentia in B partes KL; potentia verò in D partes |
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4638 <pb n=73> |
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4639 MN: tres igitur potentiæ æquales in BDE totum $u$tinebunt pon |
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4640 dus G; & vnaquæq; potentia in BD duplum $u$tinebit eius, quod |
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4641 $u$tinet potentia in E. Cùm itaq; potentia in E partem H $u$ti- |
|
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4642 neat, quæ quinta e$t pars ponderis G, ip$iq; $it æqualis; erit po |
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4643 tentia in E $ubquintupla ponderis G. & quoniam potentia in B |
|
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4644 partes kL $u$tinet, quæ quidem duplæ $unt potentiæ B, & partis H; |
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4645 erit quoq; potentia in B ip$i H æqualis: quare $ubquintupla erit |
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4646 ponderis G. Non aliter o$tendetur potentiam in D $ubquintu- |
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4647 plam e$$e ponderis G. vnaquæq; igitur potentia in BDE $ubquin- |
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4648 tupla e$t ponderis G. quod demon$trare oportebat. |
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4649 <p>Si verò $int tres vectes AB |
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4650 CD EF bifariam diui$i in |
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4651 GHk, quorum fulcimenta |
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4652 $int ACE; & pondus L eo |
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4653 dem modo in GHk $it ap- |
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4654 pen$um; quatuorq; $int po- |
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4655 tentiæ æquales in BDFG |
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4656 pondus L $u$tinentes; $imili |
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4657 modo o$tendetur vnam- |
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4658 quamq; potentiam in BD |
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4659 FG $ub$eptuplam e$$e ponde |
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4660 ris L. & $i quatuor e$$ent vectes, & quinq; potentiæ æquales pon- |
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4661 dus $u$tinentes; eodem quoq; modo o$tendetur vnamquamq; |
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4662 potentiam $ubnonuplam e$$e ponderis. atq; ita deinceps. |
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4663 <fig> |
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4664 <head>PROPOSITIO VIIII.</head> |
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4665 <p>Si quatuor duarum trochlearum binis orbi- |
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4666 culis, quarum altera $upernè, altera vero in- |
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4667 fernè, ponderiq; alligata, di$po$ita fuerit, cir |
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4668 cumducatur funis; altero eius extremo inferiori |
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4669 <foot>T</foot> |
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4670 <pb> |
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4671 trochleæ religato, altero verò à potentia pon- |
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4672 dus $u$tinente retento: erit potentia ponderis |
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4673 $ubquintupla. |
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4674 <p>Sit pondus A, cui alligata $it trochlea duos |
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4675 habens orbiculos, quorum centra $int BC; |
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4676 $itq; trochlea $ur$um appen$a duos alios ha- |
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4677 bens orbiculos, quorum centra $int DE; funi$q; |
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4678 per omnes circumducatur orbiculos, qui tro- |
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4679 chleæ inferiori religetur in F; $it qué poten |
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4680 tia in G $u$tinens pondus A. dico poten- |
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4681 tiam in G $ubquintuplam e$$e ponderis A. |
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4682 ducantur Hk LM per centra BC horizon- |
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4683 ti æquidi$tantes, quas eodem modo, quo $u- |
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4684 pra dictum e$t, e$$e tanquam vectes o$tende- |
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4685 mus, quorum fulcimenta kM, & pondus A |
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4686 ex medio vtriu$q; vectis BC $u$pen$um, & tres |
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4687 potentiæ in LHC pondus $u$tinentes, quas |
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4688 $imili modo æquales e$$e demon$trabimus; fu |
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4689 nes enim idem efficiunt, ac $i e$$ent potentiæ. |
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4690 & quoniam pondus æqualiter ex vtroq; ve- |
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4691 cte HK LM ponderat, quod quidem o$ten- |
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4692 detur quoque, vt in præcedentibus demon- |
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4693 <marg>8 <I>Huius.</I></marg> $tratum e$t: erit vnaquæq; potentia, tùm in |
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4694 L, $eu in G, quod idem e$t; tùm in H, atq; |
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4695 in C, hoc e$t in F, $ubquintupla ponderis A. |
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4696 Potentia ergo in G $u$tinens pondus A ip$ius |
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4697 A $ubquintupla erit. quod o$tendere opor- |
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|
4698 tebat. |
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|
4699 <fig> |
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|
4700 <pb n=74> |
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4701 <p>Si verò funis in F adhuc de- |
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4702 feratur circa alium orbiculum, |
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4703 cuius centrum N, qui religetur |
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4704 in O; $imiliter duplici medio |
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4705 (vt in $eptima huius) demon |
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4706 $trabitur potentiam in G pon- |
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4707 dus A $u$tinentem $ub$excu <marg><I>Ex</I> 6 <I>huius</I></marg> |
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4708 plam e$$e ponderis A. Primùm |
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4709 quidem ex tribus vectibus LM |
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4710 Hk FP, quorum fulcimenta |
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4711 $unt MkP, & pondus in me |
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4712 dio vectium appen$um; & tres |
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4713 potentiæ in LHF æquales pon |
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4714 dus $u$tinétes. deinde ex poten <marg><I>Ex</I> 8 <I>huius</I></marg> |
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4715 tiis in LHN, quarum vnaquæq; |
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4716 $ubquintupla e$$et ponderis A. |
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4717 e$$ent enim ambæ $imul poten |
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4718 tiæ in LH $ubduplæ $exquialte |
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4719 ræ ip$ius ponderis, pot&etilde;tia verò |
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4720 in F $ubdecupla e$$et, cùm $it ip |
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4721 $ius N $ubdupla: $ed duæ quin |
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4722 tæ cùm decima dimidium ef |
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4723 ficiunt, quòd $i per terna diui |
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4724 datur, $exta pars ponderis re |
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4725 $pondebit vnicuiq; potentiæ in |
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4726 LHF. ex quibus patet poten |
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4727 tiam in G $ub$excuplam e$$e |
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4728 ponderis A. $imiliterq; demon |
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4729 $trabitur vnumquemque orbi |
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4730 culum æqualem $u$tinere por- |
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4731 tionem. |
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4732 <fig> |
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4733 <foot>T 2</foot> |
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4734 <pb> |
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4735 <p>Quòd $i, vt in tertia figura |
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4736 funis in O protrahatur; per |
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4737 aliumq; circumducatur orbi- |
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4738 culum, cuius centrum Q; qui |
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4739 deinde in R trochleæ relige- |
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4740 tur inferiori; erit potentia in |
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4741 <marg><I>Ex</I> 8 <I>Huius</I></marg> G ponderis $ub$eptupla. atq; |
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4742 ita in infinitum procedendo |
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4743 proportio potentiæ ad pon- |
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4744 dus quotcunq; $ubmulti- |
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4745 plex inueniri poterit. dein- |
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4746 de $emper o$tendetur vt in |
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4747 præcedentibus; $i potentia |
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4748 pondus $u$tinens fuerit, vel |
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4749 $ubquadrupla, vel $ubquitu- |
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4750 pla, vel quouis alio modo $e |
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4751 habebit ad pondus; $imiliter |
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4752 vnumquemque funem, vel |
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4753 quartam, vel quintam, vel |
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4754 quamuis aliam partem $u$ti- |
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4755 nere ponderis, quemadmo- |
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4756 dum potentia ip$a; funes e- |
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4757 nim idem efficiunt, ac $i tot |
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4758 e$$ent potentiæ: orbiculi ve |
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4759 rò, ac $i tot e$$ent vectes. |
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4760 <fig> |
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4761 <head>COROLLARIVM</head> |
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4762 <p>Ex his manife$tum e$t orbiculos trochleæ, cui |
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4763 e$t alligatum pondus, efficere, vt pondus mino- |
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4764 <pb n=75> |
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4765 re $u$tineatur potentia, quàm $it ip$um pondus; |
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4766 quod quidem trochleæ $uperioris orbiculi non |
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4767 efficiunt. |
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4768 <p>Noui$$e tamen oportet, quòd (vt fieri $olet) inferioris tro |
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4769 chleæ orbiculus, cuius centrum N, minor e$$e debet eo, cuius cen |
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4770 trum C; hic autem minor adhuc eo, cuius centrum B; ac deniq; |
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4771 $i plures fuerint orbiculi in trochlea inferiori ponderi alligata, $em |
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4772 per cæteris maior e$$e debet, qui annexo ponderi e$t propinquior. |
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4773 oppo$ito autem modo di$ponendi $unt in trochlea $uperiori. quod |
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4774 fieri con$ueuit, ne funes inuicem complicentur; nam quantùm |
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4775 ad orbiculos attinet, $iue magni fuerint, $iue parui, nihil refert; |
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4776 cùm $emper idem $equatur. |
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4777 <p>Præterea notandum e$t, quod etiam ex dictis facilè patet, $i |
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4778 funis, $iue religetur in R trochleæ inferiori, $iue in S, maximam |
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4779 indè oriri differentiam inter potentiam, & pondus: nam $i relige |
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4780 tur in S, erit potentia in G ponderis $ub$excupla. $i verò in R, |
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4781 $ub$eptupla. quod trochleæ $uperiori non contingit, quia $iue |
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4782 religetur funis (vt in præcedenti figura) in T, $iue in O; $em |
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4783 per potentia in G $ub$excupla erit ip$ius ponderis. |
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4784 <p>Po$t hæc con$iderandum e$t, quonam modo vis moueat pon |
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4785 dus; necnon potentiæ mouentis, ponderi$q; moti $patium, atque |
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4786 tempus. |
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4787 <head>PROPOSITIO X.</head> |
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4788 <p>Si funis orbiculo trochleæ $ur$um appen$æ |
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4789 fuerit circumuolutus, cuius altero extremo $it al |
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4790 ligatum pondus; alteri autem mouens collocata |
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4791 $it potentia: mouebit hæc vecte horizonti $em- |
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|
4792 per æquidi$tante. |
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4793 <pb> |
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4794 <p>Sit pondus A, $it orbiculus trochleæ $ur |
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4795 $um appen$æ' cuius centrum K; $it deinde |
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|
4796 funis HBCDEF al igatus ponderi A in H, |
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4797 orbiculoq; circumductus; $itq; trochlea ita in |
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4798 L appen$a, & nullum alium habeat motum |
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4799 præter liberam orbiculi circa axem ver$ionem; |
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4800 $itq; potentia in F mouens pondus A. Dico |
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4801 potentiam in F $emper mouere pondus A |
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4802 vecte horizonti æquidi$tante. ducatur BKE |
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4803 horizonti æquidi$tans; $intq; BE puncta, vbi |
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4804 funes BH, & EF circulum tangunt; erit BkE |
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4805 <marg>1 <I>Huius.</I></marg> vectis, cuius fulcimentum e$t in eius medio |
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4806 k. $icut $upra o$ten$um e$t. dum itaq; vis |
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4807 in F deor$um tendit ver$us M, vectis EB |
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4808 mouebitur, cùm totus orbiculus moueatur, |
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4809 <fig> |
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4810 hoc e$t circumuertatur. dum igitur F e$t in M, $it punctum E ve |
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4811 ctis v$q; ad I motum; B autem v$q; ad C, ita vt vectis $it in |
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4812 CI. fiat deinde NM æqualis ip$i FE: & quando punctum E |
|
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4813 erit in I, tnnc funis punctum, quod erat in E, erit in N: quod au |
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4814 tem erat in B erit in C; ita vt ducta CI per centrum K tran$eat. |
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4815 dum autem B e$t in C, $it punctum H in G; eritq; BH ip$i |
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4816 CBG æqualis; cùm $it idem funis. & quoniam dum EF tendit |
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4817 in NM, adhuc $emper remanet EFM horizonti perpendicularis, |
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4818 circulumq; tangens in puncto E; ita vt ducta à puncto E per cen |
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4819 trum k, $it $emper horizonti æquidi$tans. quod idem euenit funi |
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4820 BG, & puncto B. dum igitur circulus, $iue orbiculus circumuer |
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4821 titur, $emper mouetur vectis EB, $emperq; adhuc remanet alius |
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4822 vectis in EB. $iquidem ex ip$ius rotulæ natura, in qua $emper |
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4823 dum mouetur, remanet diameter ex B in E (quæ vectis vicem ge |
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4824 rit) euenit, vt recedente vna, $emper altera $uccedat; eiu$modi |
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4825 durante circumductione: atq; ita fit, vt potentia $emper moueat |
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4826 pondus vecte EB horizonti æquidi$tante. quod demon$trare opor- |
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4827 tebat. |
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4828 <pb n=76> |
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4829 <p>Ii$dem po$itis, $patium potentiæ pondus |
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4830 mouentis e$t æquale $patio eiu$dem ponderis |
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4831 moti. |
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4832 <p>Quoniam enim o$ten$um e$t, dum F e$t in M, pondus A, hoc |
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4833 e$t punctum H e$$e in G; & cùm funis HBCDEF $it æqualis |
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4834 GBCDENFM, e$t enim idem funis; dempto igitur communi |
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4835 GBCDENF, erit HG ip$i FM æqualis. $imiliterq; o$tende- |
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4836 tur, de$cen$um F iemper æqualem e$$e a$cen$ui H. ergo $patium |
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4837 potentiæ æquale e$t $patio ponderis. quod erat demon$tran- |
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4838 dum. |
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4839 <p>Præterea potentia idem pondus per æquale |
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4840 $patium in æquali tempore mouet, tàm fune |
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4841 hoc modo orbiculo trochleæ $ur$um appen$æ |
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4842 circumuoluto, quàm $ine trochlea: dummo- |
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4843 do ip$ius potentiæ lationes in velocitate $int æ- |
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4844 quales. |
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4845 <pb> |
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4846 <p>Ii$dem po$itis $it aliud pondus P |
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4847 æquale ponderi A, cui alligatus $it |
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4848 funis TQ horizõti perp&etilde; dicularis; |
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4849 et $it TQ ip$i HB æqualis; moueat |
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4850 qué pot&etilde;tia in Q põdus P $ur$um |
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4851 ad rectos angulos horizonti, quem |
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4852 admodum mouetur pondus A. di |
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4853 co per æquale $patium in eodem |
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4854 tempore potentiam in Q pondus |
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4855 P, & potentiam in F pondus A |
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4856 mouere. quod idem e$t, ac $i e$$et |
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4857 idem pondus in æquali tempore |
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4858 motum; $icut propo$uimus. Pro- |
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4859 ducatur EF in S, & TQ in R; |
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4860 fiantq; QR FS non $olum inter |
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4861 $e $e, verùm etiam ip$i BH æqua |
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4862 les. Cùm autem TQ QR $int |
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4863 ip$is HB FS æquales, & vis in Q |
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4864 moueat pondus P per rectam T |
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4865 QR; vis autem in F moueat A |
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4866 per rectam HB, & velocitates |
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|
4867 <fig> |
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4868 motuum vtriu$q; potentiæ $int æquales; tunc in eodem tempore |
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4869 potentia in Q erit in R, & potentia in F erit in S; cùm $patia $int |
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4870 æqualia. $ed dum potentia in Q e$t in R, pondus P, hoc e$t |
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4871 punctum T erit in Q; cùm TQ $it ip$i QR æqualis. & dum po |
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4872 tentia in F e$t in S, pondus A, hoc e$t punctum H erit in B; $ed |
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4873 $patium TQ æquale e$t $patio HB, potentiæ ergo in FQ æquali |
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4874 ter motæ pondera PA æqualia per æqualia $patia in eodem tempo |
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|
4875 re mouebunt. quod erat demon$trandum |
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4876 <head>PROPOSITIO XI.</head> |
|
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4877 <p>Si funis orbiculo trochleæ ponderi alligatæ |
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4878 fuerit circumuolutus, qui in altero eius extre- |
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|
4879 <pb n=77> |
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4880 mo alicubi religetur, altero autem à potentia |
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4881 mouente pondus appræhen$o; vecte $emper ho |
|
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4882 rizonti æqui$tante potentia mouebit. |
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4883 <p>Sit pondus A; Sit orbiculus. |
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4884 CED trochleæ ponderi A alli- |
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4885 gatæ ex kH; $itq; KH ad rectos |
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4886 angulos horizonti, ita vt pon- |
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4887 dus $emper trochleæ motum, $i- |
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4888 ue $ur$um, $iue deor$um factum |
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4889 $equatur; $itq; orbiculi centrum |
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4890 K; & funis orbiculo circumuo- |
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4891 lutus $it BCDEF, qui relige- |
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4892 tur in B, ita vt in B immobilis |
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4893 maneat; & $it potentia in F mo- |
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4894 uens pondus A. dico potentia m |
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4895 in F $emper mouere põdus A ve |
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4896 cte horizonti æquidi$tante. $int |
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|
4897 BC EF inter $e $e, ip$iq; kH æ- |
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4898 quidi$tantes, & eiu$dem kH ho |
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4899 rizonti perpendiculares, tangen |
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4900 te$q; circulũ CED in EC pũctis; |
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4901 et connectatur EC, quæ per cen <marg><I>Ex</I> 1 <I>huius</I></marg> |
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4902 trum k tran$ibit, horizontiq; |
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4903 æquidi$tans erit; $icuti prius di |
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4904 ctum e$t. Quoniam enim or |
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4905 biculus CED circa eius cen |
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4906 trum K vertitur; ideo dum vis |
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4907 in F trahit $ur$um punctum E, |
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4908 deberet punctum C de$cende |
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4909 re, ac trahere deor$um B; $ed fu |
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4910 <fig> |
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4911 nis in B e$t immobilis, & BC de$cedere non pote$t; quare dum |
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4912 potentia in F trahit $ur$um E, totus orbiculus $ur$um mouebitur; |
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4913 ac per con$equens tota trochlea, & pondus; & EkC erit tanquam <marg><I>Ex</I> 2 <I>huius</I></marg> |
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4914 vectis, cuius fulcimentum erit C; e$t enim punctum C propter BC |
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4915 ferè immobile, potentia verò mouens vectem e$t in F fune EF, |
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4916 <foot>V</foot> |
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4917 <pb> |
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4918 & pondus in k appen$um. |
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4919 quòd $i punctum C omnino fue |
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4920 rit immobile, moueaturq; ve |
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4921 ctis EC in NC; & diuidatur |
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4922 NC bifariam in L: erunt CL |
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4923 LN ip$is Ck KE æquales. |
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4924 quare $i vectis EC e$$et in CN, |
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4925 punctum k e$$et in L; & $i du |
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4926 catur LM horizonti perpendi |
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4927 cularis, quæ $it etiam æqualis |
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4928 kH; e$$et pondus A, hoc e$t |
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4929 punctum H in M. $ed quoniam |
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4930 potentia in F dum tendit $ur- |
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4931 $um mouendo orbiculum, $em |
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4932 per mouetur $uper rectam EFG, |
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4933 quæ $emper e$t quoq; æquidi |
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4934 $tans BC; nece$$e erit orbicu |
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4935 lum trochleæ $emper inter li- |
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4936 neas EG BC e$$e: & centrum |
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4937 k, cum $it in medio, $uper |
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4938 rectam lineam HkT $emper |
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4939 moueri. Itaq; ducatur per L li |
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4940 nea PTLQ horizonti, & EC |
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4941 æquidi$tans, quæ $ecet Hk pro- |
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4942 ductam in T; & centro T, $pa |
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4943 tio verò TQ, circulus de$criba |
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4944 <fig> |
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4945 tur QRPS, qui æqualis erit circulo CED; & puncta PQ tangent fu |
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4946 <marg><I>Ex</I> 34 <I>primi.</I></marg> nes FE BC in PQ punctis. rectangulum enim e$t PECQ, & |
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4947 PT TQ ip$is EK kC $unt æquales. deinde per T ducatur R |
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4948 TS diameter circuli PQS æquidi$tans ip$i NC; fiatqué TO æqua |
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4949 lis kH. dum autem centrum k motum erit v$q; ad lineam PQ, |
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4950 tunc centrum k erit in T. o$ten$um e$t enim centrum orbiculi $u |
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4951 per rectam HT $emper moueri. idcirco vt centrum k $it in li |
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4952 nea PQ ip$i EC æquidi$tante, nece$$e e$t vt $it in T. & vt vectis |
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4953 EC eleuetur in angulo ECN, nece$$e e$t, vt $it in RS, non au- |
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4954 <marg>29 <I>Primi.</I></marg> tem in CN: angulus enim RSE angulo NCE e$t æqualis, & $ic |
|
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|
4955 <pb n=78> |
|
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4956 fulcimentum C non e$t penitus immobile. cùm totus orbiculus $ur |
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4957 $um moueatur, toru$q; mutet totum locum; habet tamen C ratio |
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4958 nem fulcimenti, quia minus mouetur C, quàm k, & E: punctum |
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4959 enim E mouetur v$q; ad R, & K v$q; ad T, punctum verò C v$q; |
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4960 ad S tantùm. quare dum centrum K e$t in T, po$itio orbiculi erit |
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4961 QR PS: & pondus A. hoc e$t punctum H erit in O; cùm TO |
|
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4962 $it æqualis kH; po$itio verò EC, $cilicet vectis moti, erit RS, po |
|
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|
4963 tentiaq; in F mota erit $ur$um per rectam EFG. eodem autem |
|
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4964 tempore, quo k erit in T, $it potentia in G: dum autem vectis EC |
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4965 hoc modo mouetur, adhuc $emper remanent GP BQ inter $e $e æ- |
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|
4966 quidi$tantes, atq; horizonti perpendiculares, ita vt vbi orbiculum |
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4967 tangunt, vt in punctis PQ; $emper linea PQ erit diameter orbi |
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4968 culi, & tanquam vectis horizonti æquidi$tans. dum igitur orbi- |
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4969 culus mouetur, & circumuertitur, $emper etiam mouetur vectis |
|
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4970 EC, & $emper remanet alius vectis in orbiculo horizonti æqui$tans, |
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4971 vt PQ; ita vt potentia in F $emper moueat pondus vecte hori |
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4972 zonti æquidi$tante, cuius fulcimentum erit $emper in linea CB; & |
|
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4973 pondus in medio vectis appen$um; potentiaq; in linea EG. quod |
|
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|
4974 erat o$tendendum. |
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|
4975 <p>Ii$dem po$itis, $patium potentiæ pondus |
|
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4976 mouentis duplum e$t $patii eiu$dem ponderis |
|
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4977 moti. |
|
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4978 <p>Cùm enim o$ten$um $it, dum k e$t in T, pondus A, hoc e$t |
|
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|
4979 punctum H e$$e in O, & in eodem etiam tempore potentiam in |
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4980 F e$$e in G: & quoniam funis BCDEF e$t æqualis funi BQS |
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4981 PG; funis enim e$t idem; & funis circa $emicirculum CDE e$t |
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4982 æqualis funi circa $emicirculum QSP; demptis igitur communi |
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4983 bus BQ, & FP; erit reliquus FG ip$is CQ, & EP $imul $umptis |
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4984 æqualis. $ed EP ip$i TK e$t æqualis, & CQ ip$i quoq; Tk æqualis, |
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4985 $unt enim Pk TC paralle logramma rectangula; quare lineæ EP |
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4986 CQ $imulip$ius Tk duplæ erunt. funis igitur FC ip$ius TK du |
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4987 plus erit. & quoniam kH e$t æqualis TO, dempto communi kO, |
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4988 erit kT ip$i HO æqualis; quare funis FG ip$ius HO duplus erit; |
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4989 <foot>V 2</foot> |
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4990 <pb> |
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4991 hoc e$t $patium potentiæ $patii ponderis duplum. quod erat |
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4992 demon$trandum. |
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4993 <p>Potentia deinde idem pondus in æquali tem- |
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4994 pore per dimidium $patium mouebit fune circa |
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4995 orbiculum trochleæ ponderi alligatæ reuoluto, |
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4996 quàm $ine trochlea; dummodo ip$ius potentiæ |
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4997 velocitates motuum $int æquales. |
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4998 <p>Sit enim (ii$dem po$i |
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4999 tis) aliud pondus V æqua |
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5000 le ponderi A, cui alligatus |
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5001 $it funis 9X; $itq; poten |
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5002 tia in X mouens pondus |
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5003 V. dico $i vtriu$q; poten |
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5004 tiæ motuum velocitates |
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5005 $int æquales, in eodem |
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5006 tempore potentiam in F |
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5007 mouere pondus A per di |
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5008 midium $patium eius, per |
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5009 quod à potentia in X mo |
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5010 uetur pondus V; quod |
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5011 idem e$t, ac $i e$$et idem |
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5012 pondus in æquali tempo |
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5013 re motum. Moueat po |
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5014 tentia in X pondus V, po |
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5015 tentiaq; perueniat in Y; |
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5016 $itq; XY æqualis ip$i FG; |
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5017 & fiat YZ æqualis X9, ita |
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5018 vt quando potentia in X |
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5019 erit in Y, $it pondus V, |
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5020 hoc e$t punctum 9 in Z. |
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5021 $ed 9 Z e$t æqualis FG, |
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5022 <fig> |
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5023 <pb n=79> |
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5024 cùm $it æqualis XY; ergo 9 Zip$ius HO dupla erit. Itaq; dum poten |
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5025 tiæ erunt in GY, pondera AV erunt in OZ. in eodem autem |
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5026 tempore erunt potentiæ in GY, ip$arum enim velocitates mo |
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5027 tuum $unt æquales; quare vis in F pondus A in eodem tempore |
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5028 mouebit per dimidium $patium eius, per quod mouetur à poten |
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5029 tia in X pondus V: & pondera $unt æqualia; Potentia ergo idem |
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5030 pondus in æquali tempore per dimidium $patium mouebit fune, |
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5031 trochleaq; hoc modo ponderi alligata, quàm $ine trochlea; dum |
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5032 modo potentiæ motuum velocitates $int æquales. quod erat de- |
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5033 mon$trandum. |
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5034 <head>PROPOSITIO XII.</head> |
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5035 <p>Si funis circa plures reuoluatur orbiculos, al- |
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5036 tero eius extremo alicubi religato, altero au- |
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5037 tem à potentia pondus mouente detento; poten |
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5038 tia vectibus horizonti $emper æquidi$tantibus |
|
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5039 mouebit. |
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5040 <pb> |
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5041 <p>Sit pondus A, $it orbiculus CED tro- |
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5042 chleæ ponderi alligatæ ex kS ad rectos an |
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5043 gulos horizonti; ita vt pondus $emper eius |
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5044 motum $ur$um, ac deor$um factum $equa- |
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5045 tur. $it deinde orbiculus circa centrum L |
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5046 trochleæ $ur$um appen$æ $itq; funis circa |
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5047 orbiculos reuolutus BCDEHMNO, |
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5048 qui religatus $it in B; $itq; vis in O mouens |
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5049 pondus A mouendo $e deor$um per OP. |
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5050 dico potentiam in O $emper mouere pon- |
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5051 dus A vectibus horizonti $emper æquidi. |
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5052 $tantibus. ducatur NH per centrum L ho |
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5053 <marg>1, <I>Et</I> 10 <I>Huius.</I></marg> rizonti æquidi$tans, quæ erit vectis orbi- |
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5054 culi, cuius centrum e$t L. ducatur deinde |
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5055 EC per centrum k $imiliter horizonti æqui |
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5056 <marg>11 <I>huius.</I></marg> di$tans, quæ etiam erit vectis orbiculi, cu- |
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5057 ius centrum e$t k. Moueatur potentia in |
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5058 O deor$um, quæ dum deor$um mouetur, ve |
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5059 ctem NH mouebit; & dum vectis moue- |
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5060 <marg>10 <I>Huius.</I></marg> tur, N deor$um mouebitur, H verò $ur- |
|
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5061 $um, vti$upra dictum e$t. dum autem H |
|
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5062 mouetur $ur$um, mouet etiam $ur$um E; & |
|
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|
5063 vectem EC, cuius fulcimentum e$t C, $ed |
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5064 fulcimentum C non pote$t mouere deor- |
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5065 $um B; ideo orbiculus, cuius centrum K, $ur |
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5066 <fig> |
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5067 $um mouebitur, & per con$equens trochlea, & pondus A; vt in |
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5068 præcedenti dictum e$t. & quoniam ob eandem cau$am in præce- |
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5069 dentibus a$signatam in HN, & EC $emper remanent vectes hori |
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5070 zonti æquidi$tantes; potentia ergo mouens pondus A $emper |
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5071 eum mouebit vectibus horizonti æquidi$tantibus. quod erat o- |
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5072 $tendendum. |
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5073 <p>Et $i funis circa plures $it reuolutus orbiculos; $imiliter o$tende- |
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5074 tur, potentiam mouere pondus vectibus horizonti $emper æqui- |
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5075 di$tantibus: & vectes orbiculorum trochleæ $uperioris $emper |
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5076 e$$e, vt HN, quorum fulcimenta erunt $emper in medio: vectes au- |
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5077 tem orbiculorum trochleæ inferioris $emper exi$tere, vt EC; quo- |
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5078 <pb n=80> |
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5079 rum fulcimenta erunt in extremitatibus vectium. |
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5080 <p>Ii$dem po$itis, $patium potentiæ duplum e$t |
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5081 $patii ponderis. |
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5082 <p>Sit motum centrum K v$q; ad centrum R; & orbiculus $it FTG. |
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5083 deinde per centrum R ducatur GF ip$i EC æquidi$tans: tangent |
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5084 funes EH CB orbiculum in GF punctis. fiat deniq; RQ æqua |
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5085 lis KS. dum igitur k erit in R; pondus A, $cilicet punctum S erit |
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5086 in Q. & dum centrum orbiculi e$t in R, $it potentia in O mota |
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5087 in P. & quoniam funis BCDEHMNO e$t æqualis funi BFT |
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5088 GHMNP; e$t enim idem funis; & FTG æqualis e$t CDE; dem |
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5089 ptis igitur communibus BF, & GHMNO, erit reliquus OP ip |
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5090 $is FCEG $imul $umptis æqualis: & per con$equens duplus kR, |
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5091 & QS & cùm OP $it $patium potentiæ motæ, & SQ $patium pon |
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5092 deris moti; erit $patium potentiæ duplum $patii ponderis. quod |
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5093 erat o$tendendum. |
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5094 <p>Præterea potentia idem pondus in æquali |
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5095 tempore per dimidium $patium mouebit fune |
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5096 circa duos orbiculos reuoluto, quorum vnus |
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5097 $it trochleæ $uperioris, alter verò $it trochleæ |
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5098 ponderi alligatæ; quàm $ine trochleis: dummo- |
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5099 do ip$ius potentiæ lationes $int æqualiter ve- |
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5100 loces |
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5101 <pb> |
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5102 <p>Ii$dem namq; po$itis, $it pon |
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5103 dus V æquale ip$i A, cui alliga- |
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5104 tus $it funis X9; $itq; pot&etilde;tia in X |
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5105 mouens põdus V; quæ dum pon |
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5106 dus mouet, perueniat in Y: fiant |
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5107 qué XY Z9 ip$i OP æquales; |
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5108 erit Z9 dupla QS. & $i vtriu$- |
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5109 que potentiæ velocitates mo- |
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5110 tuum $int æquales; patet pon- |
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5111 dus V duplum pertran$ire $pa- |
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5112 tium in eodem tempore eìus, |
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5113 quod pertran$it pondus A. in eo |
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5114 dem enim tempore potentia in |
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5115 X peruenit ad Y, & potentia in |
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5116 O ad P; ponderaq; $imiliter in |
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5117 ZQ. quod erat demon$tran- |
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5118 dum. |
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5119 <fig> |
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5120 <head>PROPOSITIO XIII.</head> |
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5121 <p>Fune circa $ingulos duarum trochlearum |
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5122 orbiculos, quarum altera $upernè, altera verò |
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5123 infernè, ponderiq; alligata fuerit, reuoluto; |
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5124 altero etiam eius extremo inferiori trochleæ re- |
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5125 <pb n=81> |
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5126 ligata, altero autem à mouente potentia deten- |
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5127 to: erit decur$um trahentis potentiæ $patium, mo |
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5128 ti ponderis $patii triplum. |
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5129 <p>Sit pondus A; $it BCD orbiculus tro |
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5130 chleæ ponderi A ex EQ $u$pen$o alligatæ; |
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5131 $itq; orbiculi centrum E; $it deinde FGH |
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5132 orbiculus trochleæ $ur$um appen$æ, cuius |
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5133 centrum k; $itq; funis LFGHDCBM |
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5134 circa omnes reuolutus orbiculos, tro- |
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5135 chleæq; inferiori in L religatus: $itq; in |
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5136 M potentia mouens. dico $patium de- |
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5137 cur$um à potentia in M, dum mouet pon |
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5138 dus, triplum e$$e $patii moti ponderis A. |
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5139 Moueatur potentia in M v$q; ad N; & |
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5140 centrum E $it motum v$q; ad O; & L v$ |
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5141 que ad P; atq; pondus A, hoc e$t pun- |
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5142 ctum Q v$q; ad R; orbiculu$q; motus, $it |
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5143 TSV. ducantur per EO lineæ ST BD |
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5144 horizonti æquidi$tantes, quæ inter $e $e |
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5145 quoq; æquidi$tantes erunt. quoniam au |
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5146 tem dum E e$t in O, punctum Q e$t in |
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5147 R; erit EQ æqualis OR, & EO ip$i QR |
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5148 æqualis; $imiliter LQ æqualis erit PR, |
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5149 & L Pip$i QR æqualis. tres igitur QR |
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5150 EO LP inter $e $e æquales erunt; quibus |
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5151 etiam $unt æquales BS DT. & quoniam fu |
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5152 nis LFGHDCBM æqualis e$t funi PF |
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5153 GHTVSN, cùm $it idem funis, & qui |
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5154 circa $emicirculum TVS e$t æqualis funi |
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5155 circa $emicirculum BCD; demptis igi |
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5156 tur communibus PFGHT' & SM; erit |
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5157 reliquus MN tribus BS LP DT $imul |
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5158 $umptis æqualis. BS verò LP DT $imul |
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5159 tripli $unt EO, & ex con$e quenti QR. |
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5160 <fig> |
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5161 <foot>X</foot> |
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5162 <pb> |
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5163 $patium igitur MN translatæ potentiæ $patii QR ponderis mo |
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5164 ti triplum erit. quod erat demon$trandum. |
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5165 <p>Tempus quoq; huius motus manife$tum e$t, eadem enim po |
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5166 tentia in æquali tempore $patio $ecundùm triplum ampliori $ine |
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5167 huiu$modi trochleis idem pondus mouebit, quàm cum ei$dem |
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5168 hoc modo accomodatis. $patium ponderis $ine trochleis moti |
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5169 æquale e$t $patio potentiæ. & hoc modo in omnibus inueniemus |
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5170 tempus. |
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5171 <head>PROPOSITIO XIIII.</head> |
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5172 <p>Fune circa tres duarum trochlearum orbicu |
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5173 los, quarum altera $upernè vnico dumtaxat, al |
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5174 tera verò in$ernè, duobus autem in$ignita or- |
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5175 biculis, ponderi&qacute;ue alligata fuerit, reuoluto; |
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5176 altero eius e$tremo alicubi religato, altero autem |
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5177 à potentia pondus mouente detento: erit decur- |
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5178 $um trahentis potentiæ $patium moti ponderis |
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5179 $patii quadruplum. |
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5180 <pb n=82> |
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5181 <p>Sit pondus A, $int duo orbiculi, quorū c&etilde; |
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5182 tra k I trochleæ ponderi alligatæ k <G>a</G>; ita vt |
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5183 pondus motum trochleæ $ur$um, & deor$um |
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5184 $emper $equatur: $it deinde orbiculus, cuius cen |
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5185 trum L, trochleæ $ur$um appenfæ in <35>; $itq; |
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5186 funis circa omnes orbiculos eircumuolutus BC |
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5187 DEFGHZMNO, religatu$q; in B; $itq; po |
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5188 tentia in O mouens pondus A. dico $patium, |
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5189 quod mouendo pertran$it potentia in O, qua- |
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5190 druplum e$$e $patii moti ponderis A. mouean |
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5191 tur orbiculi trochleæ ponderi alligatæ; & dum |
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5192 centrum k e$t in R, centrum I $it in S, & pon |
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5193 dus A, hoc e$t punctum <G>a</G> in <G>b</G>: erunt IS kR |
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5194 <G>ab</G> inter $e $e æquales, itemq; k Iip$i RS e- |
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5195 rit æqualis. orbiculi enim inter $e $e eandem |
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5196 $emper $eruant di$tantiam; & k <G>a</G> ip$i R <G>b</G> æ- |
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5197 qualis erit. ducantur per orbiculorum centra |
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5198 lineæ FH QT EC VX NZ horizonti æqui |
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5199 di$tantes, quæ tangent funes in FHQTEC |
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5200 VX NZ punctis, & inter $e $e quoq; æquidi |
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5201 $tantes erunt: & EQ CT VN XZ non $o |
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5202 lum inter $e $e, $ed etiam ip$is IS KR <G>ab</G> æqua |
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5203 les erunt. & dum centra kI $unt in RS, po |
|
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5204 tentia in O $it mota in P. & quoniam funis |
|
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5205 BCDEFGHZMNO e$t æqualis funi BT9 |
|
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5206 QFGHXYVP, e$t enim id&etilde; funis, & funes cir |
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|
5207 <fig> |
|
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5208 ca T9Q XYV $emicir culos $unt æquales funibus, qui $unt circa |
|
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5209 CDE ZMN; Demptis igitur communibus BT, QF GHX, |
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5210 & VO; erit OP æqualis ip$is VN XZ CT QE $imul $umptis. |
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5211 quatuor verò VN ZX CT QE $unt inter$e $e æquales, & $imul |
|
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5212 quadruplæ kR, & <G>ab</G>; quare OP quadrupla erit ip$ius <G>ab</G>. $pa |
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5213 tium igitur potentiæ quadruplum e$t $patii ponderis. quod erat |
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5214 o$tendendum. |
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5215 <p>Et $i funis in P circa alium adhuc reuoluatur orbiculum ver$us |
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5216 <35>, potentiaqué mouendo $e deor$um moueat $ur$um pondus; $imi |
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5217 liter o$tendetur $patium potentiæ quadruplum e$$e $patii ponderis. |
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5218 <foot><I>X</I> 2</foot> |
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5219 <pb> |
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5220 <p>Si verò funis in Bcircumuoluatur al |
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5221 teri orbiculo, qui deinde trochleæ in- |
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5222 <marg>9 <I>Huius.</I></marg> feriori religetur; erit potentia in O |
|
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5223 $u$tinens pondus A $ubquintupla pon |
|
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5224 deris. & $i in O $it potentia mouens |
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5225 pondus A; $imiliter demon$trabitur |
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5226 $patium potentiæ in O quintuplum e$ |
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5227 $e $patii ponderis A. |
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5228 <fig> |
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5229 <p>Et $i funis ita circa orbiculos apte- |
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5230 tur, vt potentia in O $u$tinens pon- |
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5231 dus $it ponderis $ub$extupla; & loco |
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5232 potentiæ $u$tinentis ponatur in O po- |
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5233 tentia mouens pondus: eodem modo |
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5234 o$tendetur $patium potentiæ $extu- |
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5235 plum e$$e $patii ponderis moti. & $ic |
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5236 procedendo in infinitum proportiones |
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5237 $patii potentiæ ad $patium ponderis |
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5238 moti quotcunq; multiplices inuenien- |
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5239 tur. |
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5240 <head>COROLLARIVM I.</head> |
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5241 <p>Ex his manife$tum e$t ita $e habere pondus |
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5242 ad potentiam ip$um $u$tinentem, $icuti $patium |
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5243 potentiæ mouentis ad $patium ponderis moti. |
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5244 <p>Vt $i pondus A quintuplum $it potentiæ in O pondus A $u$ti- |
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5245 nentis; erit & $patium OP potentiæ pondus mouentis quintuplum |
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5246 $patii <G>ab</G> ponderis moti. |
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5247 <pb n=83> |
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5248 <head>COROLLARIVM II.</head> |
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5249 <p>Patet etiam per ea, quæ dicta $unt, orbiculos |
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5250 trochleæ, quæ ponderi e$t alligata, efficere; vt à |
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5251 moto pondere minus, quàm à trahente poten- |
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5252 tia de$cribatur $patium; maioriq; tempore datum |
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5253 æquale $patium de$cribi, quàm $ine illis. quod |
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5254 quidem orbiculi trochleæ $uperioris non effi- |
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5255 ciunt. |
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5256 <p>Multiplici o$ten$a ponderis ad potentiam proportione, iam ex |
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5257 aduer$o potentiæ ad pondus proportio multiplex o$tendatur. |
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5258 <head>PROPOSITIO XV.</head> |
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5259 <p>Si funis orbiculo trochleæ à potentia $ur$um |
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5260 detentæ fuerit circumuolutus; altero eius extre- |
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5261 mo alicubi religato, alteri verò pondere appen |
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5262 $o; dupla erit ponderis potentia. |
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5263 <pb> |
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5264 <p>Sit trochlea habens orbiculum, cuius |
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5265 centrum A; & $it pondus B alligatum fu |
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5266 ni CDEFG, qui circa orbiculum $it re- |
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5267 uolutus, ac tandem religatus in G: $itq; |
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5268 potentia in H $u$tinens pondus. dico po |
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5269 tentiam in H duplam e$$e ponderis B. du |
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5270 catur DF per centrũ A horizonti æquidi |
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5271 $tans. quoniãigitur potentia in H $u$tinet |
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5272 trochleã, quæ$u$tinet orbiculũin eius c&etilde;tro |
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5273 A, qui pondus $u$tinet; erit potentia $u$ti |
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5274 nens orbiculũ, ac $i in A cõ$tituta e$$et; ip$a |
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5275 ergo in A exi$tente, pondere verò in D |
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5276 appen$o, funiq; CD religato; erit DF |
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5277 tanquam vectis, cuius fulcimentum erit |
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5278 F, pondus in D, & potentia in A. po- |
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5279 <marg>3 <I>Huius. de vecte.</I></marg> tentia verò ad pondus e$t, vt DF ad |
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5280 ad FA, & DF dupla e$t ip$ius FA; Po- |
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|
5281 <fig> |
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5282 tentia igitur in A, $iue in H, quodidem e$t, ponderis B dupla erit. |
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5283 quod demon$trare oportebat. |
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5284 <p>Præterea con$iderandum occurrit, cùm hæc omnia maneant, |
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5285 idem e$$e vnico exi$tente fune CD EFG hoc modo orbiculo cicum |
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5286 uoluto, ac$i duo e$$ent funes CD FG in vecte $iue libra DF al- |
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5287 ligati. |
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5288 <head>ALITER.</head> |
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5289 <p>Ii$dem po$itis, $i in G appen$um e$$et pondus k æquale pon- |
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5290 deri B, pondera B k æqueponderabunt in libra DF, cuius centrum |
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5291 A. potentia verò in H $u$tinens pondera Bk e$t ip$is $imul $um |
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5292 ptis æqualis, & pondera BK ip$ius B $unt dupla; potentia ergo in |
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5293 H ponderis B dupla erit. & quoniam funis religatus in G nihil a- |
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5294 liud efficit, ni$i quòd pondus B $u$tinet, ne de$cendat; quod idem |
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5295 efficit pondus k in G appen$um: potentia igitur in H $u$tinens |
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5296 pondus B, fune religato in G, dupla e$t ponderis B. quod de- |
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5297 mon$trare oportebat. |
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|
5298 <pb n=84> |
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|
5299 <head>PROPOSITIO XVI.</head> |
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5300 <p>Ii$dem po$itis $i in H $it potentia mouens pon |
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5301 dus, mouebit hæc eadem vecte horizonti $em- |
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5302 per æquidi$tante |
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5303 <p>Hoc etiam ($icut in $uperioribus dictum |
|
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5304 e$t) o$tendetur. moueatur enim orbiculus |
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5305 $ur$um, po$itionemq; habeat MNO, cuius |
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5306 centrum L: & per L ducatur MLO ip$i DF, |
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5307 & horizontiæquidi$tans. & quoniam funes |
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5308 tangunt circulum MON in punctis MO; |
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5309 ideo cùm potentia in A, $eu in H, quod |
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5310 idem e$t, moueat pondus B in D appen$um |
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5311 vecte DF, cuius fulcimentum e$t F; $emper |
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5312 adhuc remanebit alius vectis, .vt MO hori |
|
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5313 zonti æquidi$tans, ita vt $emper potentia |
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5314 moueat pondus vecte horizonti æquidi$tan |
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5315 te, cuius fulcimentum e$t $emper in linea |
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5316 OG, & pondus in MC, potentiaq; in cen |
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5317 tro orbiculi. |
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5318 <fig> |
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5319 <p>Ii$dem po$itis, $patium ponderis moti duplum |
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5320 e$t $patii potentiæ mouentis. |
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5321 <pb> |
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5322 <p>Sit motus orbiculus à centro A |
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5323 v$q; ad centrum L; & pondus B, |
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5324 hoce$t punctum C, in eodem tem- |
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5325 pore$it motum in P; & potentia in |
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5326 H v$q; ad K; erit AH ip$i LK æqua |
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5327 lis, & AL ip$i Hk. & quoniam fu |
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5328 nis CDEFG e$t æqualis funi PM |
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5329 NOG, idem enim e$t funis, & fu |
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5330 nis circa $emicirculum MNO æ- |
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5331 qualis e$t funi circa $emicirculum |
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5332 DEF; demptis igitur communi- |
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5333 bus DP FG, erit PC æqualis |
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5334 DM FO $imul $umptis, qui funes |
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5335 $unt dupli ip$ius AL, & con$equen- |
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5336 ter ip$ius Hk. $patium ergo pon |
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5337 deris moti CP duplum e$t $patii |
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5338 Hk potentiæ. quod oportebat de- |
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5339 mon$trare. |
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5340 <fig> |
|
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5341 <head>COROLLARIVM</head> |
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5342 <p>Exhoc manife$tum e$t, idem pondus trahi |
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5343 ab eadem potentia in æquali tempore per du- |
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5344 plum $patium trochlea hoc modo accommoda |
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5345 ta, quàm $ine trochlea; dummodo ip$ius poten |
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5346 tiæ lationes in velocitate $int æquales. |
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5347 <p>Spatium enim ponderis moti $ine trochlea æquale e$t $patio |
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5348 potentiæ. |
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|
5349 <pb n=85> |
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5350 <p>Si autem funis in G circa alium reuoluatur |
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5351 orbiculum, cuius centrum k; $itq; huiu$mo |
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5352 di orbiculi trochlea deor$um affixa, quæ nul |
|
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|
5353 lum alium habeat motum, ni$i liberam orbi |
|
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5354 culi circa axem reuolutionem; funi$q; relige |
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5355 tur in M; erit potentia in H $u$tinens pondus |
|
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5356 B $imiliter ip$ius ponderis dupla. quod qui |
|
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5357 dem manife$tum e$t, cùm idem pror$us $it, |
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5358 $iue funis $it religatus in M, $iue in G. orbicu |
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5359 lus enim, cuius centrum k, nihil efficit; penitu$ |
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|
5360 qué inutilis e$t. |
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|
5361 <fig> |
|
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5362 <p>Si verò $it potentia in M $u$tinens pon |
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5363 dus B, & trochlea $uperior $it $ur$um appen |
|
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5364 $a; erit potentia in M æqualis ponderi B. |
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5365 <p>Quoniam enim potentia in G $u$tinens <marg>1 <I>Huius.</I></marg> |
|
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5366 pondus B æqualis e$t ponderi B, & ip$i po |
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|
5367 tentiæ in G æqualis e$t potentia in L; e$t |
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|
5368 enim GL vectis, cuius fulcimentum e$t k; |
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5369 & di$tantia Gk di$tantiæ kL e$t æqualis; |
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5370 erit igitur potentia in L, $iue (quod idem e$t) |
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5371 in M, ponderi B æqualis. |
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5372 <p>Huiu$modi autem motus fit vectibus DF LG, quorum fulci |
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5373 menta $unt kA, & pondus in D, & potentia in F. $ed in vecte |
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5374 LG potentia e$t in L, pondus verò, ac $i e$$et in G. |
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5375 <p>Si deinde in M $it potentia mouens pondus, transferaturq; po |
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5376 tentia in N, pondus autem motum fuerit v$q; ad O; erit MN |
|
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5377 $patium potentiæ æquale $patio CO ponderis. Cùm enim funis |
|
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|
5378 MLGFDC æqualis $it funi NLGFDO. e$t enim idem funis; |
|
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5379 dempto communi MLGFDO; erit $patium MN potentiæ æ- |
|
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5380 quale $patio CO ponderis. |
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5381 <p>Et $i funis in M circa plures reuoluatur orbiculos, $emper erit |
|
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5382 potentia altero eius extremo pondus $u$tinens æqualis ip$i ponderi. |
|
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5383 $patiaq; ponderis, atq; potentiæ mouentis $emper o$tendentur |
|
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|
5384 æqualia. |
|
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|
5385 <foot>Y</foot> |
|
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|
5386 <pb> |
|
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5387 <head>PROPOSITIO XVII.</head> |
|
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5388 <p>Si vtri$q; duarum trochlearum $ingulis orbicu |
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5389 lis, quarum vna $upernè à potentia $u$tineatur, |
|
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|
5390 altera verò infernè, ibiq; affixa, con$tituta fue- |
|
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5391 rit, funis circumducatur; altero eius extremo $u |
|
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5392 perioritrochleæ religato, alteri verò pondere |
|
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5393 appen$o; tripla erit ponderis potentia. |
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5394 <p>Sit orbiculus, cuius centrum A, tro- |
|
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|
5395 chleæ infernè affixæ; & $it funis BCD |
|
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5396 EFG non $olum huic orbiculo circumuo |
|
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5397 lutus, verùm etiam orbiculo trochleæ $u- |
|
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5398 perioris, cuius centrum k; $itq; funis in |
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5399 B $uperioritrochleæ religatus; & in G $it ap |
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5400 pen$um pondus H; potentiaq; in L $u$ti |
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5401 neat pondus H. dico potentiam in L tri- |
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5402 plam e$$e ponderis H. $i enim duæ e$$ent |
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5403 potentiæ pondus H $u$tidentes, vna in |
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5404 K, altera in B, erunt vtræq; $imul triplæ |
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5405 <marg>15 <I>Huius. In præcedenti.</I></marg> ponderis H potentia enim in k dupla e$t |
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5406 ponderis H, & potentia in B ip$i ponderi |
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5407 æqualis. & quoniam $ola potentia in L |
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5408 vtri$q; $cilicet potentiæ in KB e$t æqua- |
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5409 lis. $u$tinet enim potentia in L; tùm po- |
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5410 tentiam in K, tùm potentiam in B; idem |
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5411 qué efficit potentia in L, ac $i duæ e$$ent |
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5412 potentiæ, vna in k, altera in B: Tri- |
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5413 pla igitur erit potentia in L ponderis H. |
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5414 quod der<*>on$trare o<*>ortebat. |
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5415 <fig> |
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5416 <pb n=86> |
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5417 <p>Si autem in L $it potentia mouens pondus. di |
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5418 co $patium ponderis moti triplum e$$e $patii po- |
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5419 tentiæ motæ. |
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5420 <p>Moueatur centrum or- |
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5421 biculi K v$q; ad M; cuius |
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5422 quidem motus $patium |
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5423 motæ potentiæ $patio e$t <marg><I>In præcedenti.</I></marg> |
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5424 æquale, $icuti $upra dictum |
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5425 e$t: & quando k erit in M, |
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5426 B erit in N; & NB æqualis |
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5427 erit M k; & dum k e$t in M, |
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5428 $it pondus H, hoc e$t pun |
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5429 ctum G motum in O; & per |
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5430 MK ducantur EF PQ ho |
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5431 rizonti æquidi$tantes; erit |
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5432 vnaquæq; EP BN FQ ip |
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5433 $i KM æqualis. & quoniam |
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5434 funis BCDEFG æqualis |
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5435 e$t funi NCDPQO; |
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5436 idem enim e$t funis; & fu- |
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5437 nis circa $emicirculum ER |
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5438 F æqualis e$t funicirca $e- |
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5439 micirculum PSQ: dem- |
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5440 ptis igitur communibus |
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5441 BCDE, & FO, erit OG |
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5442 tribus QF NB PE $imul |
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5443 $umptis æqualis. $ed QF |
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5444 NB PE $imul triplæ $unt |
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5445 Mk, hoc e$t $patii poten- |
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5446 tiæ motæ; $patium ergo |
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5447 GO ponderis H moti tri- |
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|
5448 <fig> |
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5449 plum e$t $patii potentiæ motæ. quod o$tendere oportebat. |
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|
5450 <foot>Y 2</foot> |
|
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|
5451 <pb> |
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|
5452 <head>PROPOSITIO XVIII.</head> |
|
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5453 <p>Si vtriu$q; duarum trochlearum binis orbicu |
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5454 lis, quarum altera $upernè à potentia $u$tineatur, |
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5455 altera verò infernè, ibiq; annexa, collocata fue- |
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5456 rit, funis circumnectatur; altero eius extremo |
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5457 alicubi, non autem $uperiori trochleæ religato, |
|
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5458 alteri verò pondere appen$o; quadrupla erit |
|
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|
5459 ponderis potentia. |
|
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|
5460 <p>Sit trochlea inferior, duos habens orbiculos, |
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5461 quorum centra AB; $it qué trochlea $uperior |
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5462 duos $imiliter habens orbiculos, quorum cen- |
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5463 tra CD; funi$q; EFGHKLMNOP $it cir- |
|
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5464 ca omnes orbiculos reuolutus, qui $it religatus |
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|
5465 in E; & in P appendatur pondus Q; $itq; po- |
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|
5466 tentia in R. dico potentiam in R quadruplam |
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5467 e$$e ponderis Q. Cùm enim $i duæ intelligan |
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5468 tur potentiæ, vna in k, altera in D, potentia |
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5469 <marg>16 <I>Huius.</I></marg> in k $u$tinens pondus Q fune k LMNOP æ- |
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5470 qualis erit ponderi; erunt duæ$imul potentiæ, |
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5471 vna in D, altera in k, pondus Q $u$tinentes, |
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5472 triplæ eiu$dem ponderis. Potentia verò in C |
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5473 dupla e$t potentiæ in k, & per con$equens pon |
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5474 deris Q; idem enim e$t, ac $i in k appen$um e$ |
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5475 <marg>15 <I>Huius.</I></marg> $et pondus æquale ponderi Q, cuius dupla e$t |
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5476 potentia in C; duæ igitur potentiæ in DC qua- |
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|
5477 druplæ $unt ponderis Q. & cùm potentia in R |
|
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5478 orbiculis $u$tineat pondus Q, erit pot&etilde;tia in R, |
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5479 ac $i duæ e$$ent potentiæ, vna in D, altera in C, |
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5480 & vtræq; $imul pondus Q $u$tinerent. ergo po- |
|
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|
5481 tentia in R quadrupla e$t ponderis Q. quod |
|
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|
5482 oport<*>bat demon$trare. |
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5483 <fig> |
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5484 <pb n=87> |
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5485 <head>COROLLARIVM</head> |
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5486 <p>Ex quo patet, $i funis fuerit religatus in G, & |
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5487 circa orbiculos, quorum centra $unt BCD reuo- |
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5488 lutus; potentiam in R pondus $u$tinentem $imili- |
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5489 ter ponderis Q quadruplam e$$e. orbiculus enim, |
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5490 cuius centrum A, nihil efficit. |
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5491 <p>Si autem in R $it potentia mouens pondus. dico |
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5492 $patium ponderis moti quadruplum e$$e $patii |
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5493 potentiæ. |
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5494 <p>Moue antur centra CD orbiculorum v$q; ad |
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5495 ST; erunt ex $uperius dictis CS DT $patio |
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5496 potentiæ æqualia; & per CSDT ducantur Hk |
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5497 VX NO YZ horizonti æquidi$tantes; & dũ |
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5498 centra CD $unt in ST, $it pondus Q, hoc e$t |
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5499 punctum P motum in 9. & quoniam funis EF |
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5500 GHKLMNOP æqualis e$t funi EFGVX |
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5501 LMYZ 9; cùm $it idem funis: & funes circa |
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5502 $emicirculos NIO H <G>a</G> k $unt æquales funi- |
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5503 bus, qui $unt circa $emicirculos Y<35>Z V<G>b</G>X; |
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5504 demptis igitur communibus EFGH kLMN |
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5505 & O9; erit P9 ip$is NY ZO VH <I>X</I>k $i- |
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5506 mul $umptis æqualis. quatuor autem NY ZO |
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5507 VH Xk $imul quadrupli $unt DT, hoc e$t |
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5508 $patii potentiæ; $patium igitur P9 ponderis |
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5509 quadruplum e$t $patii potentiæ quod demon |
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5510 $trandum fuerat. |
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5511 <fig> |
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5512 <pb> |
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5513 <p>Si autem funis $it re- |
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5514 ligatus in E trochleæ $u |
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|
5515 periori, & potentia in R |
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5516 $u$tineat pondus Q; e- |
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5517 rit potentia in R ponde |
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5518 ris Q quintupla. & $i in |
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5519 R $it potentia mouens |
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5520 pondus; erit $patium pon |
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5521 deris moti quintuplum |
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5522 $patii potentiæ. quæ om- |
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5523 nia $imili modo o$ten- |
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5524 dentur, $icut in præce- |
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|
5525 dentibus demon$tra- |
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|
5526 tum e$t. |
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|
5527 <fig> |
|
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|
5528 <pb n=88> |
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5529 <p>Si verò potentia in R $ub$tineat pon- |
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5530 dus Q trochlea tres orbiculos habente, |
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|
5531 quorum centra $int ABC; & $it alia tro |
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|
5532 chlea infernè af$ixa duos, vel tres orbicu- |
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|
5533 los habens, quorum centra DEF; $itq; |
|
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5534 funis circa omnes orbiculos reuolutus, $i- |
|
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|
5535 ue in G, $iue in H religatus; $imiliter |
|
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|
5536 o$tendetur potentiam in R $excuplam |
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|
5537 e$$e ponderis Q. Et $i in R $it potentia |
|
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5538 mouens pondus, o$tendetur $patium pon |
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|
5539 deris moti $excuplum e$$e $patii poten- |
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|
5540 tiæ. |
|
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|
5541 <fig> |
|
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5542 <p>Et $i funis $it religatus in K trochleæ |
|
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|
5543 $uperiori, & in R $it potentia pondus |
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5544 $u$tinens; $imili modo o$tendetur poten |
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|
5545 tiam in R $eptuplam e$$e ponderis Q. |
|
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5546 <p>Et $i in R $it potentia mouens, o$ten |
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|
5547 detur $patium ponderis Q $eptuplum e$$e |
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|
5548 $patii potentiæ. atq; ita in infinitum |
|
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|
5549 omnis potentiæ ad pondus multiplex |
|
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|
5550 proportio inueniri poterit. $emperq; o- |
|
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|
5551 $tendetur, ita e$$e pondus ad potentiam |
|
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|
5552 ip$um $u$tinentem, $icuti $patium poten |
|
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|
5553 tiæ pondus mouentis ad $patium ponde- |
|
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|
5554 ris moti. |
|
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|
5555 <p>Vectium autem ip$orum orbiculorum |
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|
5556 motus in his fit hoc modo, videlicet vectes |
|
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|
5557 orbiculorum trochleæ $uperioris mouen |
|
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|
5558 tur, vti dictum e$t in decima $exta huius; |
|
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|
5559 hoc e$t habent fulcimentum in extremita |
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|
5560 te, potentiam in medio, pondus in altera extremitate appen$um. ve |
|
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|
5561 ctes verò trochleæ inferioris habent fulcimentum in medio, pon |
|
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|
5562 dus, & potentiam in extremitatibus. |
|
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|
5563 <pb> |
|
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|
5564 <head>COROLLARIVM</head> |
|
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|
5565 <p>Manife$tum e$t in his, orbiculos trochleæ $u |
|
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|
5566 perioris efficere, vt pondus moueatur maiori |
|
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|
5567 potentia, quàm $it ip$um pondus, & per maius |
|
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|
5568 $patium potentiæ $patio, & per æquale tempo- |
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5569 re minori; quod quidem orbiculi trochleæ in- |
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5570 ferioris non efficiunt. |
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5571 <p>Alio quoq; modo hanc potentiæ ad pondus multiplicem propor |
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5572 tionem inuenire po$$umus. |
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5573 <head>PROPOSITIO XVIIII.</head> |
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5574 <p>Si vtriu$q; duarum trochlearum $ingulis orbi |
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5575 culis, quarum altera $upernè appen$a, altera ve- |
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5576 rò infernè à $u$tinente potentia rententa fuerit, |
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5577 funis circumuoluatur; altero eius extremo alicu |
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5578 bi religato, alteri autem pondere appen$o; du- |
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5579 pla erit ponderis potentia. |
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5580 <pb n=89> |
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5581 <p>Sit orbiculus trochleæ $upernè appen$æ, cu |
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5582 ius centrum $it A; & BCD $it trochleæ infe |
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5583 rioris; $it deinde funis EBC DFGHL reli- |
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5584 gatus in E; & in L $it appen$um pondus M; |
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5585 $itq; potentia in N $u$tinens pondus M. |
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5586 dico potentiam in N duplam e$$e ponderis |
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5587 M. Cùm enim $upra o$ten$um $it potentiam |
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5588 in L, quæ pondus, exempli gratia, O $u$ti- <marg>3 <I>Huius.</I></marg> |
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5589 neat in N appen$um, $ubduplam e$$e eiu$dem |
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5590 ponderis; potentia igitur in N ponderi O æ- |
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5591 qualis pondus M potentiæ in L æquale $u$ti |
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5592 nebit; ponderi$q; M dupla erit. quod demon |
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5593 $trare oportebat. |
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5594 <fig> |
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5595 <head>ALITER.</head> |
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5596 <p>Ii$dem po$itis. Quoniam potentia in F, <marg>1 <I>Huius.</I></marg> |
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5597 $eu in D, quod idem e$t, æqualis e$t ponde |
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5598 ri M; & BD e$t vectis, cuius fulcimentum |
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5599 e$t B, & potentia in N e$t, ac $i e$$et in me- |
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5600 dio vectis, & pondus æquale ip$i M, ac$i e$- |
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5601 $et in D propter funem FD; quod idem |
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5602 e$t, ac $i BCD e$$et orbiculus trochleæ $upe |
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5603 rioris, pondusq; appen$um e$$et in fune DF, |
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5604 $icut in decimaquinta, & decima$exta dictum e$t; ergo potentia in |
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5605 N dupla e$t ponderis M. quod erat o$tendendum. |
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5606 <p>Si autem in N $it potentia mouens pondus M, erit $patium |
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5607 ponderis M duplum $patii potentiæ in N. quod ex duodecima |
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5608 huius manife$tum e$t; $patium enim puncti L deor$um ten- |
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5609 dentis duplum e$t $pat^{1}i N $ur$um; erit igitur è conuer$o $patium |
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5610 potentiæ in N deor$um tendentis dimidium $aptii ponderis M $ur |
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5611 $um moti. |
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5612 <p>Sicut autem ex tertia, quinta, $eptima huius, &c. colligi po$$unt |
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5613 ponderis O rationes quotcunq; multiplices ip$ius potentiæ in L, |
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5614 eod&etilde; quoq; modo o$tendi poterunt potentiæ in N pondus $u$tinen |
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5615 tis ponderis M quotcunq; multiplices. Atq; ita ex decimatertia |
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5616 <foot>Z</foot> |
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5617 <pb> |
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5618 decimaquarta rationes o$ten |
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5619 dentur quotcunq; multiplices |
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5620 $patii ponderis M ad $patium |
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5621 potentiæ mouentis in N con$ti |
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5622 tutæ. |
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5623 <fig> |
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5624 <p>Poterit quoq; ex decima$e |
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5625 ptima decimaoctaua huius mul |
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5626 tiplex inueniri proportio, quam |
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5627 habet potentia pondus $u$ti |
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5628 nens ad ip$um pondus; $icut |
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5629 proportio potentiæ in N ad pon |
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5630 dus M ex decimaquinta, & deci |
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5631 ma$exta o$tendebatur: inuenie |
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5632 turq; ita e$$e pondus ad poten |
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5633 tiam pondus $u$tinentem, vt $pa |
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5634 tium potentiæ mouentis ad $pa |
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5635 tium ponderis. |
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5636 <p>Vectium motus in his fit |
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5637 hoc modo, videlicet vectes or |
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5638 biculorum trochleæ inferioris |
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5639 mouentur, vt vectis BD, quæ |
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5640 mouetur, ac$i B e$$et fulcimen |
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5641 tum, & pondus in D, & poten |
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5642 tia in medio. Vectes verò or |
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5643 biculorum trochleæ $uperioris mouentur, vt FH, cuius fulcimen |
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5644 tum e$t in medio, pondus in H, & potentia in F. |
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5645 <head>COROLLARIVM.</head> |
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5646 <p>Ex hoc manife$tum e$t, orbiculos trochleæ |
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5647 inferioris in his efficere, vt pondus maiori po- |
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5648 <pb n=90> |
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5649 tentia moueatur, quàm $it ip$um pondus, & |
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5650 per maius $patium $patio potentiæ, & minori |
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5651 tempore per æquale. quod quidem orbiculi $u |
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5652 perioris trochleæ non efficiunt. |
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5653 <p>Cognitis proportionibus multiplicibus, iam ad $uperparticu |
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5654 lares accedendum e$t. |
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5655 <head>PROPOSITIO XX.</head> |
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5656 <p>Si vtriu$q; duarum trochlearum $ingulis or- |
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5657 biculis, quarum altera $upernè à potentia $u$ti- |
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5658 neatur, altera verò infernè, ponderiq; alligata, |
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5659 cõ$tituta fuerit, funis reuoluatur; altero eius extre |
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5660 mo alicuibi, altero verò inferiori trochleæ reli |
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5661 gato; pondus potentiæ $e$quialterum erit. |
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5662 <foot>Z 2</foot> |
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5663 <pb> |
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5664 <p>Sit ABC orbiculus |
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5665 trochleæ $uperioris, & |
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5666 DEF trochleæ inferio- |
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5667 ris ponderi G alligatæ; |
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5668 $itq; funis HABCDE |
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5669 Fk circa orbiculos re- |
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5670 uolutus, qui $it religatus |
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5671 in K, & in H trochleæ |
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5672 inferiori; $itq; potentia |
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5673 in L $u$tinens pondus |
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5674 G. dico pondus poten |
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5675 tiæ $e$quialterum e$$e. |
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5676 <marg><I>Cor.</I> 5 <I>huius.</I></marg> Quoniam enim vterque |
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5677 funis CD AH tertiam |
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5678 $u$tinet partem ponde- |
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5679 ris G, erit vnaquæq; po |
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|
5680 tentia in DH $ubtripla |
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5681 ponderis G; quibus $i- |
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5682 mul a$$umptis e$t æqua- |
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|
5683 <fig> |
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5684 <marg><I>Ex.</I> 15 <I>huius.</I></marg> lis potentia in L: potentia enim in L dupla e$t potentiæ in D, & |
|
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5685 eius, quæ e$t in H. quare potentia in L $ub$e$quialtera e$t ponde- |
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5686 ris G. pondus ergo G ad pontentiam in L e$t, vt tria ad duo; |
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5687 hoc e$t $e$quialterum. quod demon$trare oportebat. |
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|
5688 <pb n=91> |
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5689 <p>Si autem in L $it potentia mouens pondus. |
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5690 Dico $patium potentiæ $patii ponderis $e$quial- |
|
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5691 terum e$$e. |
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5692 <p>Ii$dem po$itis, perueniat orbi- |
|
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5693 culus ABC v$q; ad MNO, & |
|
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|
5694 DEF ad PQR; & H in S; & |
|
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|
5695 pondus G v$q; ad T. Et quoniam |
|
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5696 funis HABCDEFK e$t æqualis |
|
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|
5697 funi SMNOPQRk, cùm $it |
|
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|
5698 idem funis; & $unes circa $emicir |
|
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|
5699 culos ABC MNO $unt inter $e |
|
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|
5700 $e æquales; qui verò $unt circa |
|
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|
5701 DEF PQR $imiliter inter $e æ- |
|
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|
5702 quales; Demptis igitur AS CP |
|
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|
5703 RK communibus, erunt duo CO |
|
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|
5704 MA tribus DP HS FR æqua- |
|
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|
5705 les. $ed vterq; CO AM $eor$um |
|
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|
5706 e$t æqualis $patio potentiæ motæ. |
|
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|
5707 quare duo CO MA, $imul $patii |
|
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|
5708 potentiæ dupli erunt: tre$q; DP |
|
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|
5709 HS FR $imul $imili modo $patii |
|
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|
5710 ponderis moti tripli erunt. dimidia |
|
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5711 verò pars, hoc e$t $patium poten |
|
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|
5712 tiæ motæ ad tertiam, ad $patium |
|
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|
5713 $cilicet ponderis moti ita $e habet, |
|
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5714 vt duplum dimidii ad duplum ter- |
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|
5715 tii; hoc e$t, vt totum ad duas ter |
|
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|
5716 <fig> |
|
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5717 tias, quod e$t vt tria ad duo. $patium ergo potentiæ in L $pa- |
|
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5718 tii ponderis G moti $e$quialterum e$t. quod o$tendere opor- |
|
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|
5719 tebat. |
|
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|
5720 <pb> |
|
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|
5721 <head>PROPOSITIO XXI.</head> |
|
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5722 <p>Si tribus duarum trochlearum orbiculis, qua |
|
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|
5723 rum altera vnius tantùm orbiculi $upernè à po- |
|
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|
5724 tentia $u$tineatur, altera verò duorum infernè, |
|
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|
5725 ponderiq; alligata, collocata fuerit, funis cir- |
|
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|
5726 cumuoluatur; altero eius extremo alicubi, altero |
|
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|
5727 autem $uperiori trochleæ religato: ponduspoten |
|
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|
5728 tiæ $e$quitertium erit. |
|
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|
5729 <p>Sit pondus A trochleæ inferiori alliga- |
|
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|
5730 tum, quæ duos habeat orbiculos, quorum |
|
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|
5731 centra $int BC; $uperiorq; trochlea orbicu- |
|
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|
5732 lum habeat, cuius centrum D; & $it funis |
|
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|
5733 EFGHkLMN circa omnes orbiculos re |
|
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|
5734 uolutus, qui religatus $it in N, & in E tro |
|
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|
5735 chleæ $uperiori; $itqué potentia in O |
|
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|
5736 $u$tinens pondus A. dico pondus po- |
|
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|
5737 <marg><I>Cor.</I> 1 <I>$eptime buius.</I></marg> tentiæ $e$quitertium e$$e. Quoniam enim |
|
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|
5738 vnu$qui$q; funis NM HG EF KL quar- |
|
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5739 tam $u$tinent partem ponderis A, & omnes |
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5740 $imul totum $u$tinent pondus; tres HG |
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5741 EF kL $imul tres $u$tinebunt partes pon- |
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5742 deris A. quare pondus A ad hos omnes |
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5743 $imul erit, vt quatuor ad tria: & cùm po- |
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5744 tentia in O idem efficiat, quod HG EF kL |
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5745 $imul efficiunt; omnes enim $u$tinet; erit po |
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5746 tentia in O tribus $imul HG EF kL æ- |
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5747 qualis; & ob id pondus A ad potentiam |
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5748 in O erit, vt quatuor ad tria; hoc e$t $e$qui |
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5749 tertium. quod demon$trare oportebat. |
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5750 <fig> |
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5751 <pb n=92> |
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5752 <p>Si vero in O $it potentia mouens pondus A. |
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5753 Dico $patium potentiæ in O decur$um $patii pon |
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5754 deris A moti $e$quitertium e$$e. |
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5755 <p>Ii$dem po$itis, $it centrum B motum |
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5756 in P; &C v$q; ad Q; & D in R; & E in |
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5757 S eodem tempore: & per centra ducantur |
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5758 ML 9Z FG TV Hk XY horizonti, |
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5759 & inter $e $e æquidi$tantes. Similiter, vt in |
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5760 præcedente o$tendetur tres <I>X</I>H SE Yk |
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5761 quatuor TG VF ZL 9M æquales e$$e. & |
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5762 quoniam tres XH SE Yk $imul triplæ |
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5763 $unt $patii potentiæ, quatuorverò TG VF |
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5764 ZL 9M $imul quadruplæ $unt $patii pon |
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5765 deris moti; erit $patium potentiæ ad $pa- |
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5766 tium ponderis, vt tertia pars ad quartam. |
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5767 $ed tertia pars ad quartam e$t, vt tres ter |
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5768 tiæ ad tres quartas, hoc e$t, vt totum ad |
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5769 tres quartas; quod e$t, vt quatuor ad tria. |
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5770 $patium ergo potentiæ $patii ponderis mo |
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5771 ti $e$quitertium e$t. quod erat demon- |
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5772 $trandum. |
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5773 <fig> |
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5774 <p>Si verò funis in E per alium circumuol |
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5775 uatur orbiculum, qui deinde trochleæ in |
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5776 feriori religetur; $imiliter o$tendetur pro |
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5777 portionem ponderis ad potentiã in O pon |
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5778 dus $u$tinentem $e$quiquartam e$$e. quòd |
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5779 $i in O $it potentia mouens pondus, o$ten |
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5780 detur $patium potentiæ $patii ponderis $e$ |
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5781 quiquartum e$$e. & $ic in infinitum proce |
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5782 dendo quamcunq; $uperparticularem pro |
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5783 portionem ponderis ad potentiam inuenie |
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5784 mus; $emperq; reperiemus, ita e$$e pondus |
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5785 ad potentiam pondus $u$tinentem, vt $pa- |
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5786 tium potentiæ mouentis ad $patium ponde- |
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|
5787 ris moti. |
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5788 <pb> |
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|
5789 <p>Motus verò vectium fit hoc mo |
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5790 do, videlicet vectis ML fulci- |
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|
5791 mentum e$t M, cùm funis $it re |
|
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5792 ligatus in N, & pondus in me- |
|
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5793 dio, & potentia in L. quia ve- |
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5794 rò punctum L tendit $ur$um, quod |
|
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5795 à fune KL mouetur, idcirco K $ur- |
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5796 $um mouebitur, & vectis HK ful |
|
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5797 cimentum erit H, pondus ac $i e$ |
|
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5798 $ent in k, & potentia in medio; |
|
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5799 vectis autem FG fulcimentum |
|
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5800 erit G, pondus in medio; & poten |
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|
5801 tia in F. punctum enim F $ur$um |
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5802 mouetur à fune EF. Præterea |
|
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5803 G in orbiculo deor$um tendit, |
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5804 quia H quoque in eius orbiculo |
|
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5805 deor$um mouetur. |
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|
5806 <fig> |
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|
5807 <pb n=93> |
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5808 <head>PROPOSITIO XXII.</head> |
|
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5809 <p>Si vtri$que duarum trochlearum $ingulis |
|
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|
5810 orbiculis, quarum altera $upernè à potentia |
|
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5811 $u$tineatur, altera verò infernè, ponderiq; alli- |
|
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5812 gata, collocata fuerit, circumducatur funis; al- |
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5813 tero eius extremo alicubi, altero autem $uperio |
|
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5814 ri trochleæ religato. erit potentia ponderis $e$ |
|
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|
5815 quialtera. |
|
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5816 <p>Sit orbiculus ABC trochleæ ponderi D al |
|
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5817 ligatæ; & EFG trochleæ $uperioris, cuius |
|
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|
5818 centrum H; $it deinde funis k ABCEFGL |
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5819 circa orbiculos reuolutus, & religatus in L, & |
|
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5820 in k trochleæ $uperiori; $itq; potentia in M |
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|
5821 $u$tinens pondus D. dico potentiam ponde |
|
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5822 ris $e$quialteram e$$e. Quoniam enim poten <marg>2 <I>Huius.</I></marg> |
|
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|
5823 tia in E $u$tinens pondus D $ubdupla e$t pon <marg><I>Ex</I> 15 <I>huius.</I></marg> |
|
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5824 deris D, potentiæ verò in E dupla e$t poten <marg>2 <I>Cor.</I></marg> |
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5825 tia in H; erit potentia in H ponderi D æqua <marg>2 <I>Huius.</I></marg> |
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5826 lis; & cùm potentia in K $ubdupla $it ponde |
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5827 ris D; erunt vtræq; $imul potentiæ in H k $e$ |
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5828 quialteræ ponderis D. Itaq; cùm potentia in |
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5829 M duabus potentiis in Hk $imul $umptis $it |
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5830 æqualis, quemadmodum in $uperioribus o- |
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5831 $ten$um e$t; erit potentia in M $e$quialtera |
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5832 ponderis D. quod oportebat demon$trare. |
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5833 <p>Si verò in M $it potentia mouens pondus, |
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5834 $imiliter vt in præcedentibus o$tendetur, $pa |
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5835 tium ponderis $patii potentiæ $e$quialterum |
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5836 e$$e. |
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5837 <fig> |
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5838 <foot>Aa</foot> |
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5839 <pb> |
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5840 <p>Et $i funis in K per alium circumuoluatur |
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5841 orbiculum, cuius centrum $it N; qui dein- |
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5842 de trochleæ inferiori religetur in O; & po- |
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5843 tentia in M $u$tineat pondus D. dico pro- |
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5844 portionem potentiæ ad pondus $e$quiter- |
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5845 tiam e$$e. |
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5846 <fig> |
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5847 <p>Quoniam enim potentia in E $u$tinens |
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5848 <marg>5 <I>Huius.</I></marg> pondus D fune ECB AKPO $ubtripla e$t |
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5849 <marg><I>Ex</I> 15 <I>huius.</I></marg> ip$ius D, ip$ius autem E dupla e$t potentia |
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5850 in H; erit potentia in H $ub$e$quialtera pon |
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5851 deris D. $imili quoq; modo quoniam po |
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5852 tentia in O_{3} quæ e$t, ac $i e$$et in centro or |
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5853 <marg>3, 15,<I>Huius.</I></marg> biculi ABC, $ubtripla e$t ponderis D; ip- |
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5854 $ius autem O dupla e$t potentia in N; erit |
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5855 quoq; potentia in N $ub$e$quialtera ponde- |
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5856 ris D. quare duæ $imul potentiæ in HN pon |
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5857 dus D $uperant tertia parte, $e $e habentq; ad |
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5858 D in ratione $e$quitertia: & cùm potentia |
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5859 in M duabus $it potentiis in HN $imul $um |
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5860 ptis æqualis, $uperabit itidem potentia in |
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5861 M pondus D tertia parte. ergo proportio |
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5862 potentiæ in M ad pondus D $e$quitertia |
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5863 e$t. quod demon$trare oportebat. |
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5864 <p>Si autem in M $it potentia mouens pon- |
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5865 dus, $imili modo o$tendetur $patium ponderis D $patii potentiæ in |
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5866 M $e$quitertium e$$e. |
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5867 <p>Et $i funis in O per alium circumuoluatur orbiçulum, qui tro- |
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5868 chleæ $uperiori deinde religetur; eodem modo demon$tr abimus |
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5869 proportionem potentiæ in M pondus $u$tinentis ad pondus $e$- |
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5870 quiquartam e$$e. & $i in M $it potentia mouens, $imiliter o$ten- |
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5871 detur $patium ponderis $patii potentiæ $e$quiquartum e$$e. pro- |
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5872 cedendoq; hoc modo in infinitum quamcunq; proportionem |
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5873 potentiæ ad pondus $uperparticularem inueniemus; $emperqué |
|
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|
5874 <pb n=94> |
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5875 o$tendemus potentiam pondus $u$tinentem ita e$$e ad pondus, |
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5876 vt $patium ponderis ad $patium potentiæ pondus mouentis. |
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5877 <p>Motus verò vectis EG e$t, ac $i G e$$et fulcimentum, cùm |
|
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5878 funis $it religatus in L; pondus ac $i in E e$$et appen$um, & po- |
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5879 tentia in medio. Vectis verò CA fulcimentum e$t A pondus in |
|
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5880 medio, & potentia in C. & K fulcimentum e$t vectis Pk, pon- |
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5881 dus in P, & potentia in medio. quæ omnia $icut in præceden- |
|
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|
5882 ti o$tendentur. |
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|
5883 <head>PROPOSITIO XXIII.</head> |
|
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5884 <p>Si vtri$q; duarum trochlearum $ingulis or- |
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5885 biculis, quarum altera $upernè à potentia $u$ti- |
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5886 neatur, altera verò infernè, ponderiq; alligata, |
|
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5887 cõ$tituta fuerit, circumferatur funis; vtroq; eius |
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|
5888 extremo alicuibi, non autem trochleis religato; |
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5889 æqualis erit ponderi potentia. |
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|
5890 <foot>Aa 2</foot> |
|
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5891 <pb> |
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5892 <p>Sit orbiculus trochleæ $uperioris |
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5893 ABC, cuius centrum D; & EFG |
|
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5894 trochleæ ponderi H alligatæ, cu- |
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5895 ius centrum k; & $it funis LEF |
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5896 GABCM circa orbiculos reuo- |
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5897 lutus, religatu$q; in LM; $itq; |
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5898 potentia in N $u$tinens pondus |
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5899 H. dico potentiam in N æqua |
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5900 lem e$$e ponderi H. Accipiatur |
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5901 quoduis punctum O in AG. & |
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5902 quoniam $i in O e$$et potentia $u |
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5903 <marg>2 <I>Huius.</I></marg> $tinens pondus H, $ubdupla e$$et |
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5904 <marg><I>Ex</I> 15 <I>huius.</I></marg> ponderis H, & potentiæ in O |
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5905 dupla e$t ea, quæ e$t in D, $iue |
|
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5906 (quod idem e$t) in N; erit po |
|
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5907 tentia in N ponderi H æqualis. |
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5908 quod demon$trare oportebat. |
|
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|
5909 <fig> |
|
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5910 <p>Et $i in N $it potentia mouens pondus. Dico |
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|
5911 $patium potentiæ in N æqualem e$$e $patio pon |
|
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|
5912 deris H moti. |
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5913 <p>Quoniam enim $patium puncti O moti, duplum e$t, tùm $patii |
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5914 <marg>11 <I>Huius.</I></marg> ponderis H moti, tùm $patii potentiæ in N motæ; erit $patium |
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5915 <marg>16 <I>Huius.</I></marg> potentiæ in N $patio ponderis H æquale. |
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|
5916 <pb n=95> |
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|
5917 <head>ALITER.</head> |
|
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5918 <p>Ii$dem po$itis, transfera |
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|
5919 tur centrum orbiculi ABC |
|
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5920 v$q; ad P; orbiculu$q; po$i |
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|
5921 tionem habeat QRS; dein |
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|
5922 de eodem tempore orbiculus |
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5923 EFG $it in TVX, cuius cen |
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5924 trum $it Y; & pondus perue |
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5925 nerit in Z. ducantur per or |
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5926 biculorum centra lineæ GE |
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5927 TX AC QS horizonti æqui |
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5928 di$tantes. & $icut in aliis |
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5929 demon$tratum fuit, d uo fu- |
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5930 nes AQ CS duobus XG |
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5931 TE æquales erunt; $ed AQ |
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5932 CS $imul dupli$unt $patii po |
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|
5933 tentiæ motæ; & duo XG TE |
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5934 $imul $unt $imiliter dupli $pa |
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5935 tii ponderis; erit igitur $patiũ |
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5936 potentiæ $patio ponderis æ- |
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5937 quale. quod demon$trare o- |
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|
5938 portebat. |
|
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|
5939 <fig> |
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|
5940 <pb> |
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5941 <p>Quod etiam $i vtraq; trochlea duos |
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|
5942 habuerit orbiculos, quorum centra |
|
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|
5943 $int ABCD, funi$q; per omnes cir |
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|
5944 cumuoluatur, qui in LM religetur; |
|
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|
5945 $imiliter o$tendetur potentiam in N |
|
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|
5946 æqualem e$$e ponderi H. vnaquæq; |
|
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5947 enim potentia in EF $u$tinens pon- |
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|
5948 dus $ubquadrupla e$t ponderis; & po |
|
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|
5949 tentiæ in CD duplæ $unt earum, |
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5950 quæ $unt in EF; erit vnaquæq; po- |
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|
5951 tentia in CD $ubdupla ponderis H. |
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|
5952 quare potentiæ in CD $imul $umptæ |
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5953 ponderi H erunt æquales. & quo- |
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|
5954 niam potentia in N duabus in CD |
|
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|
5955 pontentiis e$t æqualis; erit potentia |
|
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|
5956 in N ponderi H, æqualis. |
|
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|
5957 <p>Et $i in N $it potentia mouens, $i |
|
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|
5958 mili modo o$tendetur, $patium po- |
|
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|
5959 tentiæ æquale e$$e $patio ponderis. |
|
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|
5960 <p>Si autem vtraq; trochlea tres, vel |
|
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|
5961 quatuor, vel quotcunq; habeat orbi- |
|
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|
5962 culos; $emper o$tendetur pot&etilde;tiam in |
|
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|
5963 N æqualem e$$e ponderi H; & $pa |
|
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|
5964 tium potentiæ pondus mouentis æ- |
|
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|
5965 quale e$$e $patio ponderis moti. |
|
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|
5966 <fig> |
|
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|
5967 <p>Vectium autem motus hoc pacto $e habent; orbiculorum qui |
|
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|
5968 dem trochleæ $uperioris, veluti AC in præcedenti figura fulcimen |
|
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|
5969 tum e$t C, pondus verò in A appen$um, & potentia in D medio. |
|
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|
5970 vectes autem orbiculorum trochleæ inferioris ita mouentur, vt ip |
|
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|
5971 $ius GE fulcimentum $it E, pondus in medio appen$um, & po |
|
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|
5972 tentia in G. |
|
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|
5973 <pb n=96> |
|
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|
5974 <head>PROPOSITIO XXIIII.</head> |
|
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|
5975 <p>Si tribus duarum trochlearum orbiculis, qua |
|
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|
5976 rum altera vnius dumtaxat orbiculi $upernè à |
|
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|
5977 potentia $u$tineatur, altera verò duorum infer- |
|
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|
5978 nè, ponderiq, alligata fuerit con$tituta, cir- |
|
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|
5979 cundetur funis; vtroq; eius extremo alicubi, $ed |
|
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|
5980 non $uperiori trochleæ religato: duplum erit |
|
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|
5981 pondus potentiæ. |
|
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|
5982 <p>Sint AB centra orbiculorum |
|
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|
5983 trochleæ ponderi C alligatæ; D ve |
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|
5984 rò $it centrum orbiculi trochleæ $u |
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|
5985 perioris; $it deinde funis per om |
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|
5986 nes orbiculos circumuolutus, reli |
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|
5987 gatu$q; in EF; & $it potentia in |
|
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|
5988 G $u$tinens pondus C. dico pon |
|
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|
5989 dus C duplum e$$e potentiæ in G. |
|
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|
5990 Quoniam enim $i in H k duæ e$- |
|
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|
5991 $ent potentiæ pondus $u$tinentes |
|
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|
5992 duobus funibus orbiculis trochleæ |
|
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5993 inferioris tantùm circumuolutis, e$ |
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5994 $et vtiq; vtraq; potentia in k H $ub <marg><I>Ex</I> 7 <I>huius</I></marg> |
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5995 quadrupla ponderis C; $ed poten- |
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5996 tia in G æqualis e$t potentiis in Hk <marg><I>Ex</I> 15 <I>huius.</I></marg> |
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5997 $imul $umptis; vniu$cuiu$q; enim |
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5998 potentiæ in H, & k dupla e$t: erit |
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5999 potentia in G $ubdupla ponderis |
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6000 C. pondus ergo potentiæ duplum |
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6001 erit. quod demon$trare opor- |
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6002 tebat. |
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6003 <fig> |
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6004 <pb> |
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6005 <p>Et $i in G $it potentia mouens pondus. Dico |
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6006 $patium potentiæ duplum e$$e $patii ponderis. |
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6007 <p>Ii$dem po$itis, $int |
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6008 moti orbiculi, $imiliter |
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6009 demon$trabitur ambos |
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6010 illos LM NO æquales |
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6011 e$$e quatuor PQ RS |
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6012 TV XY. $ed LM NO |
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6013 $imul dupli$unt $patii po |
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6014 tentiæ in G motæ; & |
|
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6015 quatuor PQ RS TV |
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6016 XY $imul quadrupli $unt |
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6017 $patii ponderis moti.$pa |
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6018 tium igitur potentiæ ad |
|
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6019 $patium ponderis e$t tan |
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6020 quam $ubduplum ad $ub |
|
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|
6021 quadruplum. erit ergo |
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6022 potentiæ $patium pon- |
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|
6023 deris $patii duplum. |
|
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|
6024 <fig> |
|
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|
6025 <pb n=97> |
|
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6026 <p>Hinc autem con$iderandum |
|
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|
6027 e$t quomodo fiat motus; quia, |
|
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|
6028 cùm funis $it religatur in F, vectis |
|
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6029 NO in prima figura habebit ful- |
|
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|
6030 cimentum O, pondus in medio, |
|
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6031 & potentia in N. $imiliter quo- |
|
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|
6032 niam funis e$t religatus in E, ve |
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6033 ctis PQ habebit fulcimentũ P, & |
|
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|
6034 pondus in medio, & potentia in |
|
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|
6035 Q. idcirco partes orbiculorum |
|
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|
6036 in N, & Q $ur$um mouebuntur; |
|
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|
6037 orbiculi ergo non in eandem, $ed |
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6038 in contrarias mouebuntur partes, |
|
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|
6039 videlicet vnus dextro$um, alter$i- |
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|
6040 ni$tror$um. & quoniam potentiæ |
|
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|
6041 in NQ eædem $unt, quæ $unt in |
|
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|
6042 LM; potentiæ igitur in LM æ- |
|
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|
6043 quales $ur$um mouebuntur. ve |
|
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|
6044 ctis igitur LM in neutram moue |
|
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|
6045 bitur partem. quare neq; orbicu |
|
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|
6046 lus circumuertetur. Itaq; LM |
|
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|
6047 erit tanquam libra, cuius centrum |
|
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|
6048 D, ponderaqué appen$a in LM |
|
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|
6049 æqualia quartæ parti ponderis C; |
|
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|
6050 vnu$qui$q; enim funis LN MQ |
|
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|
6051 quartam $u$tinet partem ponderis C. mouebitur ergo totus orbi |
|
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|
6052 culus, cuius centrum D, $ur$um; $ed non circumuertetur. |
|
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|
6053 <fig> |
|
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|
6054 <foot>Bb</foot> |
|
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|
6055 <pb> |
|
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|
6056 <p>Et $i funis in F circa alios duos |
|
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|
6057 voluatur orbiculos, quorum cen- |
|
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|
6058 tra $int HK, qui deinde religetur |
|
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|
6059 in L; erit proportio ponderis ad |
|
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|
6060 potentiam $e$quialtera. |
|
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|
6061 <p>Si enim quatuor e$$ent potentiæ |
|
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|
6062 <marg><I>Ex</I> 9 <I>huius</I></marg> in MNOI, e$$et vnaquæq; $ub$e$- |
|
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|
6063 cupla ponderis C, quare quatuor |
|
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|
6064 $imul potentiæ in MNOI qua- |
|
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|
6065 tuor $extæ crunt ponderis C. & |
|
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|
6066 quoniam duæ $imul potentiæ in |
|
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|
6067 HD quatuor potentiis in MNOI |
|
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|
6068 $unt æquales; & potentia in G æ- |
|
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|
6069 qualis e$t potentiis in DH: erit |
|
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|
6070 potentia in G quatuor $imul po- |
|
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|
6071 tentiis in MNOI æqualis; & ob |
|
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|
6072 id quatuor $extæ erit ponderis C. |
|
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|
6073 proportio igitur ponderis C ad po |
|
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|
6074 tentiam in G $e$quialtera e$t. |
|
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|
6075 <p>Et $i in G $it potentia mouens, |
|
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|
6076 $imili modo o$tendetur $patium |
|
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|
6077 potentiæ $patii ponderis $e$quialte |
|
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|
6078 rum e$$e. |
|
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|
6079 <fig> |
|
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6080 <p>Et $i funis in L adhuc circa duos |
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6081 alios orbiculos reuoluatur $imi- |
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6082 liter o$tendetur proportionem |
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6083 ponderis ad potentiam $e$qui- |
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6084 tertiam e$$e. quòd $i in G $it |
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6085 potentia mouens, o$tende- |
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6086 tur $patium potentiæ $patii ponde |
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6087 ris $e$quitertium e$$e, atq; ita dein- |
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6088 ceps in infinitum procedendo, |
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6089 quamcunq; proportionem ponderis ad potentiam $uperparticula |
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6090 rem inueniemus $emperq; reperiemus ita e$$e pondus ad poten |
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6091 tiam pondus $u$tinentem, vt $patium potentiæ mouentis ad $pa |
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6092 tium ponderis à potentia moti. |
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6093 <pb n=98> |
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6094 <p>Motus vectium fit hoc modo, vectis YZ, cùm funis $it religatus |
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6095 in E, habet fulcimentum in Y, pondus in B medio appen$um, & |
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6096 potentia in Z. & vectis PQ habet fulcimentum in P potentia in |
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6097 medio, & pondus in Q. oportet enim orbiculos, quorum cen- |
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6098 tra$unt BD in eandem partem moueri, videlicet vt QZ $ur- |
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6099 $um moueantur. & quoniam funis religatus e$t in L, erit T fulci |
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6100 mentum vectis ST, qui pondus habet in medio, & potentia in |
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6101 S. & quia S mouetur $ur$um, nece$$e e$t etiam R $ur$um moue |
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6102 ri; & ideo F erit fulcimentum vectis FR, & pondus erit in R, |
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6103 & potentia in medio. orbiculi igitur, quorum centra $unt H k, |
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6104 in contrariam mouentur partem eorum, quorum centra $unt BD: |
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6105 quare partes orbiculorũ PF in orbiculis deor$um tend&etilde;t; videlicet |
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6106 ver$us XV. vectis igitur VX in neutram partem mouebitur, cùm |
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6107 P, & F deor$um moueantur; & VX erit tanquam vectis, in cuius |
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6108 medio erit pondus appen$um, & in VX duæ potentiæ æquales |
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6109 $extæ parti ponderis C. potentiæ enim in MO hoc e$t funes PV |
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6110 FX $extam $u$tinent partem ponderis C. totus igitur orbiculus, |
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6111 cuius centrum A $ur$um vnà cum trochlea mouebitur; non au- |
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6112 tem circumuertetur. |
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6113 <head>PROPOSITIO XXV.</head> |
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6114 <p>Si tribus duarum trochlearum orbiculis, |
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6115 quarum altera binis in$ignita rotulis à potentia |
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6116 $upernè detineatur; altera verò vnius tantùm |
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6117 rotulæ infernè cõ$tituta, ac ponderi alligata fue |
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6118 rit, circumuoluatur funis; vtroq; eius extremo |
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6119 alicuibi, non autem inferiori trochleæ religa- |
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6120 to: dupla erit ponderis potentia. |
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6121 <foot>Bb 2</foot> |
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6122 <pb> |
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6123 <p>Sit pondus A trochleæ inferiori alligatum, |
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6124 quæ orbiculum habeat, cuius centrum $it B; tro |
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6125 chlea verò $uperior duos orbiculos habeat, |
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6126 quorum centra $int CD; $itq; funis circa om |
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6127 nes orbiculos reuolutus, qui in EF $it religatus; |
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6128 potentiaq; $u$tinens pondus $it in G. dico po |
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6129 tentiam in G ponderis A duplam e$$e. $i enim |
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6130 <marg>2. <I>Cor.</I></marg> in H k duæ e$$ent potentiæ pondus $u$tinen |
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6131 <marg>2 <I>Huius.</I></marg> tes, e$$et vtraq; $ubdupla ponderis A; $ed po |
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6132 <marg><I>Ex</I> 15 <I>huius.</I></marg> tentia in D dupla e$t potentiæ in H, & poten |
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6133 tia in C dupla potentiæ in K; quare duæ $imul |
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6134 potentiæ in CD vtriu$q; $imul potentiæ in H k |
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6135 duplæ erunt. $ed potentiæ in H k ponderi A $unt |
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6136 æquales, & potentiæ in CD ip$i potentiæ in G |
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6137 $unt etiam æquales; potentia igitur in G ponde- |
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6138 ris A dupla erit. quod oportebat demon$trare. |
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6139 <p>Si autem in G $it potentia mouens pon- |
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6140 dus, $imiliter vt in præcedenti o$tendetur $pa |
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6141 tium ponderis $patii potentiæ duplum e$$e. |
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6142 <fig> |
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6143 <p>Hinc quoq; con$iderandum e$t vectem PQ |
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6144 non moueri, quia vectis LM habet fulcimen |
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6145 tum in L, potentia in medio, & pondus in M. |
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6146 vectis autem NO habet fulcimentum in O, |
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6147 potentia in medio, & pondus in N. quare M, & N $ur$um mo |
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6148 uebuntur. in contrarias igitur partes orbiculi, quorum centra |
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6149 $unt CD mouentur. idcirco vectis PQ in neutram partem mo |
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6150 uebitur; eritq;, ac $i in medio e$$et appen$um pondus, & in PQ |
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6151 duæ potentiæ æquales dimidio ponderis A. vtraq; enim potentia |
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6152 in HK $ubdupla e$t ponderis A. totus igitur orbiculus, cuius |
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6153 centrum B $ur$um mouebitur, $ed non circumuertetur. |
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6154 <pb n=99> |
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6155 <p>Et $i funis in F duobus aliis adhuc circumuol- |
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6156 uatur orbiculis, quorum centra $int HK, qui de- |
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6157 inde religetur in L; erit proportio potentiæ in G |
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6158 ad pondus A $e$quialtera. |
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6159 <p>Si enim in MNOP quatuor e$$ent poten |
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6160 tiæ pondus $u$tinentes, vnaquæq; $ubquadru <marg><I>Ex</I> 7 <I>huius</I></marg> |
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6161 pla e$$et ponderis A: $ed cùm potentia in k <marg>15 <I>Huius.</I></marg> |
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6162 $it dupla potentiæ in N; erit potentia in k |
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6163 ponderis A $ubdupla. & quoniam potentia |
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6164 in D duabus in MO potentiis e$t æqualis; erit |
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6165 quoq; potentia in D ponderis A $ubdupla. |
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6166 cùm autem adhuc potentia in C potentiæ in P |
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6167 $it dupla, erit $imiliter pot&etilde;tia in C ponderis A |
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6168 $ubdupla. tres igitur potentiæ in CD k tribus |
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6169 medietatibus ponderis A $unt æquales. quo- |
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6170 niam autem potentia in G potentiis in CDK |
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6171 e$t æqualis, erit potentia in G tribus medie- |
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6172 tatibus ponderis A æqualis. Proportio igi- |
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6173 tur potentiæ ad pondus $e$quialtera e$t. |
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6174 <p>Si verò in G $it potentia mouens, erit $pa |
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6175 tium ponderis $patii potentiæ $e$quialterum. |
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6176 <fig> |
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6177 <p>Et $i funis in L adhuc circa duos alios or |
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6178 biculos reuoluatur, $imiliter o$tendetur pro- |
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6179 portionem potentiæ ad pondus $e$quitertiam |
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6180 e$$e. & $ic in infinitum omnes proportiones |
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6181 potentiæ ad pondus $uperparticulares inue- |
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6182 niemus. o$tendemu$q; potentiam pondus |
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6183 $u$tinentem ad pondus ita e$$e, vt $patium |
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6184 ponderis moti ad $patìum potentiæ pondus |
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6185 mouentis. |
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6186 <pb> |
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6187 <p>Motus vectium fiet hoc |
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6188 modo, videlicet Q erit ful |
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6189 cimentum vectis QR, po- |
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6190 tentia in medio, pondus |
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6191 in R; & vectis Z 9 fulci |
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6192 mentum erit Z, pondus in |
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6193 medio, potentiaq; in 9. $i |
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6194 militer X erit fulcimentum |
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6195 vectis VX, potentia in me |
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6196 dio, pondus in V. & quo |
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6197 niam V $ur$um mouetur, Y |
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6198 quoq; $ur$um mouebitur; |
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6199 & vectis YF fulcimentum |
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6200 erit F: quare F, & Z in orbi |
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6201 culis deor$um mouebun- |
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6202 tur. & ob id vectis ST in |
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6203 neutram mouebitur par- |
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6204 tem; & ST erit tamquam |
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6205 libra, cuius centrum D, & |
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6206 pondera in ST æqualia |
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6207 quartæ parti ponderis A. |
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6208 vnu$qui$q; enim funis SZ |
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6209 TF quartam $u$tinet par- |
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6210 tem ponderis A. orbicu- |
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6211 lus ergo, cuius centrum D, |
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6212 $ur$um mouebitur; non au |
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6213 tem circumuertetur. |
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6214 <fig> |
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6215 <pb n=100> |
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6216 <p>Hactenus proportiones ponderis ad potentiam multiplices, |
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6217 & $ubmultiplices; deinde $uperparticulares, $ub$uperparticu- |
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6218 lare$qué declaratæ fuerunt: nunc autem reliquum e$t, vt propor- |
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6219 tiones inter pondus, & potentiam $uperpartientes, & multi- |
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6220 plices $uperparticulares, multiplicesqué $uperpartientes mani- |
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6221 fe$tentur. |
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6222 <head>PROPOSITIO XXVI.</head> |
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6223 <head>PROBLEMA.</head> |
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6224 <p>Si proportionem $uperpartientem inuenire |
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6225 volumus, quemadmodum $i proportio, quam |
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6226 habet pondus ad potentiam pondus $u$tinen- |
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6227 tem fuerit $uperbipartiens, $icut quinque ad |
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6228 tria. |
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6229 <pb> |
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6230 <p><marg><I>Ex</I> 9 <I>huius.</I></marg> Exponatur potentia in A pondus B $u$ti |
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6231 nens, proportionemq; habeat pondus B ad |
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6232 potentiam in A, vt quinq; ad vnum; hoc e$t, |
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6233 $it potentia in A $ubquintupla ponderis B: de- |
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6234 inde eodem fune circa alios orbiculos reuo- |
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6235 <marg><I>Ex</I> 17 <I>huius.</I></marg> luto inueniatur potentia in C, quæ tripla $it |
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6236 potentiæ in A. & quoniam pondus Bad po |
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6237 tentiam in A e$t, vt quinq; ad vnum; & |
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6238 potentia in A ad potentiam in C e$t, vt vnum |
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6239 ad tria; erit pondus B ad potentiam in C, vt |
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6240 quinq; ad tria; hoc e$t $uperbipartiens. |
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6241 <p>Et hoc modo omnes proportiones ponde |
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6242 ris ad potentiam $uperpartientes inuenientur; |
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6243 vt $i $upertripartientem quis inuenire volue- |
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6244 rit; eodem incedat ordine; fiat $cilicet poten |
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6245 tia in A $u$tinens pondus B $ub$eptupla ip- |
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6246 $ius ponderis B; deinde fiat potentia in C ip- |
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6247 $ius A quadrupla; erit pondus B ad poten- |
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6248 tiam in C, vt $eptem ad quatuor: vídelicet |
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6249 $upertripartiens. |
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6250 <p>Si verò in C $it potentia mo- |
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6251 uens pondus erit $patium pot&etilde;tiæ |
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6252 $patii ponderis $uperbipartiens. |
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6253 <fig> |
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6254 <p><marg>17 <I>Huius.</I></marg> Spatium enim potentiæ in C tertia pars |
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6255 e$t $patii potentiæ in A, ita videlicet $e habent, |
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6256 vt quinq; ad quindecim; & $patium potentiæ |
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6257 <marg>14 <I>Huius.</I></marg> in A quintuplum e$t $patii ponderis B, hoc |
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6258 e$t, vt quindecim ad tria; erit igitur $patium |
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6259 potentiæ in C ad $patium ponderis B, vt |
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6260 quinq; ad tria; videlicet $uperbipartiens. & $emper o$tendemus, ita |
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6261 e$$e $patium potentiæ mouentis ad $patium ponderis; vt pondus |
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6262 ad potentiam pondus $u$tinentem. |
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6263 <p>Similiq; pror$us ratione proportionem potentiæ ad pondus $u- |
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6264 <pb n=101> |
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6265 perpartientem inueniemus. $i enim C e$$et inferius, & in ip$o |
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6266 appen$um e$$et pondus; B verò $uperius, in quo e$$et potentia pon |
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6267 dus in C $u$tinens, e$$et potentia in B $uperbipartiens ponderis |
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6268 in C appen$i: cùm B ad A $it, vtquinq; ad vnum; A verò ad <marg>18 <I>Huius.</I></marg> |
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6269 C, vt vnum ad tria. <marg>5 <I>Huius.</I></marg> |
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6270 <p>Si autem multiplicem $uperparticularem in- |
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6271 uenire voluerimus; vt proportio, quam habet |
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6272 pondus ad potentiam pondus $u$tinentem, $it |
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6273 duplex $e$quialtera, vt quinq; ad duo. |
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6274 <p>Eodem modo, quo $uperpartientes inuenimus, has quo- |
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6275 que omnes multiplices $uperparticulares reperiemus. vt fiat <marg><I>Ex</I> 9 <I>huius.</I></marg> |
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6276 pondus B ad potentiam in A, vt quinq; ad vnum; potentia ve <marg><I>Ex</I> 15, 16, <I>Huius.</I></marg> |
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6277 rò in C ad potentiam in A, vt duo ad vnum; quod fiet, $i fu- |
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6278 nis $it religatus in D, non autem trochleæ $uperiori, velin F: erit |
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6279 pondus B ad potentiam in C, vt quinq; ad duo; hoc e$t duplum |
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6280 $e$quialterum. |
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6281 <p>Et è conuer$o proportionem potentiæ ad pondus multiplicem |
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6282 $uperparticularem inueniemus; & vt in reliquis o$tendetur, ita e$ |
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6283 $e $patium potentiæ mouentis ad $patium ponderis, vt pondus |
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6284 ad potentiam pondus $u$tinentem. |
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6285 <p>Omnem quoq; multiplicem $uperpartientem |
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6286 eodem modo inueniemus; vt $i proportio, quam |
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6287 habet pondus ad potentiam, $it duplex $uperbi |
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6288 partiens, vt octo ad tria. |
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6289 <p>Fiat potentia in A pondus B $u$tinens $uboctupla ponderis B; <marg><I>Ex</I> 9 <I>huius Ex</I> 17 <I>huius.</I></marg> |
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6290 & potentia in C potentiæ in A $it tripla; erit pondus B ad po |
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6291 tentiam in C, vt octo ad tria. & è conuer$o omnem potentiæ ad |
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6292 <foot>Cc</foot> |
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6293 <pb> |
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6294 pondus proportionem multipticem $uperpartientem in ueniemus. |
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6295 & vt in cæteris reperiemus ita e$$e pondus ad potentiam pondus |
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6296 $u$tinentem, vt $patium potentiæ mouentis ad $patium pon- |
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6297 deris. |
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6298 <p>Notandum autem e$t, quòd cùm in præcedentibus demo$tratio |
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6299 nibus $æpius dictum fuerit, potentiam pondus $u$tinentem ip$ius |
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6300 ponderis duplam e$$e, vel triplam, & huiu$modi; vt in decima- |
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6301 quinta huius o$ten$um e$t; quia tamen potentia non $olum pon |
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6302 dus, verùm etiam trochleam $u$tinet; idcirco maioris longè vir- |
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6303 tutis, maiori$q; ip$i ponderi proportionis con$tituenda videtur |
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6304 ip$a potentia. quod quidem verum e$t, $i etiam trochleæ graui |
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6305 tatem con$iderare voluerimus. $ed quoniam inter potentiam, & |
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6306 pondus proportionem quærimus: ideo hanc trochleæ grauit at em |
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6307 ommi$imus, quam $iquis etiam con$iderare voluerit, vim ip$i po- |
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6308 tentiæ æqualem trochleæ addere poterit. Quod ip$um etiam in |
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6309 fune ob$eruari poterit. & $icut hoc in decimaquinta con$ideraui |
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6310 mus, idem quoq; in reliquis aliis con$iderare poterimus. |
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6311 <pb n=97> |
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6312 <p>Noui$$e etiam oportet, quòd $icuti proportio |
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6313 nes omnes inter potentiam, & pondus vnico |
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6314 fune inuentæ fuerunt; ita etiam pluribus funi- |
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6315 bus, trochlei$qué eædem inueniri poterunt. vt |
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6316 $i multiplicem $uperparticularem proportionem |
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6317 pluribus funibus inuenire voluerimus, veluti $i |
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6318 proportio, quam habet pondus ad potentiam |
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6319 pondus $u$tinentem, fuerit duplex $e$quialtera, vt |
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6320 quinq; ad duo; oportet hanc proportionem ex |
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6321 pluribus componere. vt (exempli gratia) ex pro- |
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6322 portione $e$quiquarta, vt quinqué ad quatuor, |
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6323 & ex dupla, vt quatuor ad duo. exponatur igitur po <marg><I>Ex</I> 21 <I>huius.</I></marg> |
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6324 tentia in A pondus B $u$tinens, ad quam pondus |
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6325 proportion&etilde; habeat $e$quiquartam, vt quinq; ad |
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6326 quatuor: deinde alio fune inueniatur pot&etilde;tia in C,<marg><I>Ex</I> 2 <I>huius.</I></marg> |
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6327 cuius dupla $it potentia in A. & quoniã B ad A e$t, |
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6328 vt quinq; ad quatuor; & A ad C, vt quatuor ad |
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6329 duo; erit pondus B ad potentiam in C, vt quin |
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6330 que ad duo; hoc e$t proportionem habebit du- |
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6331 plicem $eíquialteram. |
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6332 <fig> |
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6333 <p>Et notandum e$t hanc quoq; proportion&etilde; inue |
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6334 niri po$$e, $i proportionem quinq; ad duo ex pluri |
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6335 bus componamus, vt quinq; ad quindecim & quin |
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6336 decim ad viginti & viginti ad duo. Et hoc modo |
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6337 non $olum omnem aliam proportionem inuenie |
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6338 mus, $ed quamcunq, multis, infinitisqué mo- |
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6339 dis comperiemus. omnis enim proportio ex infi- |
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6340 nitis proportionibus componi pote$t. vt patet |
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6341 in commentario Eutocii in quartam propo$itio- |
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6342 nem $ecundi libri Archimedis de $phera, & cy- |
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6343 lindro. |
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6344 <p>Po$$umus quoq; pluribus funibus, trochleis |
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6345 verò inferioribus tantùm, vel $uperioribus vti. |
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6346 <foot>Cc 2</foot> |
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6347 <!-- |
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6348 <pb> |
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6349 --> |
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6350 <pb> |
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6351 <p>Sit pondus A, cui alligata $it trochlea |
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6352 orbiculum habens, cuius centrum B; |
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6353 religetur funis in C, qui circa orbiculum |
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6354 reuoluatur, funi$q; perueniat in D: erit |
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6355 <marg>2 <I>Huius.</I></marg> potentia in D $u$tinens pondus A $ub- |
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6356 dupla ponderis A. deinde funis in D |
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6357 alteri trochleæ religetur, & circa huius |
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6358 trochleæ orbiculum alius reuoluatur fu |
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6359 nis, qui religetur in E, & perueniat in |
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6360 <marg>2 <I>Huius.</I></marg> F; erit potentia in F $ubdupla eius, |
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6361 quod $u$tinet pot&etilde;tia in D: e$tenim ac$i |
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6362 D dimidium ponderis A $u$tineret $i |
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6363 ne trochlea; quare potentia in F $ubqua- |
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6364 drupla erit ponderis A. & $i adhuc fu |
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6365 nis in F alteri trochleæ religetur, & |
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6366 per eius orbiculum circumuoluatur a- |
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6367 lius funis, qui religetur in G, & per |
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6368 ueniat in H; erit potentia in H $ub |
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6369 dupla potentiæ in F. ergo potentia in |
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6370 H $uboctupla erit ponderis A. & $ic |
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6371 in infinitum $emper $ubduplam poten |
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6372 tiam præced&etilde;tis potentiæ inueniemus. |
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6373 <fig> |
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6374 <p>Et $i in H $it potentia mouens, erit |
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6375 $patium potentiæ $patii ponderis octu |
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6376 <marg>11 <I>Huius.</I></marg> plum. $patium enim D duplum e$t $pa |
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6377 tii ponderis A, & $patium F $patii D |
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6378 duplum; erit $patium F $patii ponde |
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6379 ris A quadruplum. $imiliter quoniam |
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6380 $patium potentiæ in H duplũ e$t $patii |
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6381 F, erit $patium potentiæ in H $patii |
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6382 ponderis A octuplum. |
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6383 <pb n=103> |
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6384 <p>Sit deinde pondus A funi alliga- |
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6385 tum, qui orbiculo trochleæ $uperio |
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6386 ris $it circumuolutus, & religatus in <marg>15 <I>Huius.</I></marg> |
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6387 B; $itq; potentia in C $u$tinens pon |
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6388 dus A: erit potentia in C ponderis A |
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6389 dupla, deinde C alterifuni religetur, |
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6390 qui per alterius trochleæ orbicu |
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6391 lum circumuoluatur, & religetur |
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6392 in D; erit potentia in E dupla po <marg><I>Ex e adem.</I></marg> |
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6393 tentiæ in C. Quare potentia in E |
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6394 quadrupla erit ponderis A. & $i ad |
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6395 huc E alteri funi religetur, qui etiam |
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6396 circa orbiculum alterius trochleæ re |
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6397 uoluatur, & religetur in F; erit poten |
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6398 tia in G dupla potentiæ in E. ergo |
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6399 potentia in G octupla erit ponderis |
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6400 A. & $ic in infinitum $emper præ |
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6401 cedentis potentiæ potentiam du- |
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6402 plam inueniemus. |
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6403 <fig> |
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6404 <p>Si autem in G $it potentia mo- <marg>16 <I>Huius.</I></marg> |
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6405 uens, erit $patium ponderis octu- |
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6406 plum $patii potentiæ in G. $patium |
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6407 enim ponder is A duplum e$t $patii |
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6408 potentiæ in C, & C duplum e$t $patii |
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6409 ip$ius E; quare $patium ponderis |
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6410 A $patii potentiæ in E quadruplum |
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6411 erit. $imiliter quoniam $patium E |
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6412 duplum e$t $patii potentiæ in G; erit ergo $patium ponderis A |
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6413 octuplum $patii potentiæ in G. |
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6414 <pb> |
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6415 <head>COROLLARIVM.</head> |
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6416 <p>Ex his manife$tum e$t maiorem $emper ha- |
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6417 bere proportionem $patium potentiæ mouen- |
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6418 tis ad $patium ponderis moti, quàm pondus |
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6419 ad eandem potentiam. |
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6420 <p>Hoc autem ex iis, quæ in corollario quartæ huius de vecte dicta |
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6421 $unt, patet. |
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6422 <head>PROPOSITIO XXVII.</head> |
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6423 <head>PROBLEMA.</head> |
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6424 <p>Datum pondus à data potentia trochleis |
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6425 moueri. |
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6426 <p>Data potentia, vel e$t maior, vel æqualis, vel minor dato |
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6427 pondere. |
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6428 <pb n=104> |
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6429 <p>Et $i e$t maior, tunc poten- |
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6430 tia, vel ab$q; alio in$trumento, |
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6431 vel fune circa orbiculum trochleæ |
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6432 $ur$um appen$æ reuoluto datum |
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6433 pondus mouebit. Minor enim po <marg><I>Ex</I> 1 <I>huius</I></marg> |
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6434 tentia; quàm data, ponderiæque- |
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6435 ponderat, data ergo mouebit. |
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6436 Quod idem fieri pote$t iuxta om- |
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6437 nes propo$itiones, quibus poten- |
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6438 tia pondus $u$tinens, vel æqualis, |
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6439 vel minor pondere o$ten$a e$t. |
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6440 <fig> |
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6441 <p>Si autem æqualis, |
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6442 pondus mouebit fune |
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|
6443 per orbiculum trochleæ |
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6444 ponderialligatæ circum |
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6445 uoluto. potentia enim <marg>2 <I>Huius.</I></marg> |
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6446 $u$tinens pondus $ubdu |
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6447 pla e$t ponderis, poten |
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6448 tia igitur ponderi æqua |
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6449 lis datum pondus mo- |
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6450 uebit. Quod etiam $e- |
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6451 cundùm propo$itiones, |
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6452 quibus potentiam pon |
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6453 dere minorem e$$e o- |
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6454 $ten$um e$t, fieri po- |
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6455 te$t. |
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|
6456 <fig> |
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|
6457 <pb> |
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6458 <p>Si verò minor, $it datum pondus |
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6459 vt $ex aginta, potentia verò mouens |
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6460 <marg><I>Ex</I> 9 <I>huius</I></marg> data $it tredecim. inueniatur poten- |
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6461 tia in A $u$tinens pondus B, quæ pon |
|
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|
6462 deris B $it $ubquintupla. & quoniam |
|
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|
6463 potentia in A pondus $u$tinens e$t |
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6464 vt duodecim; maior igitur poten- |
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|
6465 tia, quàm duodecim in A pondus |
|
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6466 B mouebit. Quare potentia vt tre- |
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|
6467 decim in A pondus B mouebit. quod. |
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|
6468 facere oportebat. |
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6469 <p>Animaduertendũ quoq; e$t in mo |
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6470 uendis ponderibus, potentiam ali- |
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6471 quando for$itan melius mouere mo |
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|
6472 uendo $e deor$um, quàm mouendo |
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|
6473 $e $ur$um. vt circumuoluatur adhuc |
|
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|
6474 funis per alium trochleæ $uperioris |
|
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|
6475 orbiculum, cuius centrum C, funi$q; |
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6476 <marg><I>Ex</I> 5 <I>Huius</I></marg> perueniat in D; erit pot&etilde;tia in D $u$ti |
|
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|
6477 n&etilde;s põdus B $imiliter duodecim, qu&etilde; |
|
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|
6478 admodum erat in A. Ideo poten- |
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|
6479 tia vt tredecim in D pondus B mo- |
|
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6480 uebit. & quia mouet $e deor$um, |
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|
6481 forta$$e trahet facilius, quàm in A; |
|
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|
6482 atq; tempus e$t idem, $icut etiam |
|
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6483 erat in A. |
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|
6484 <fig> |
|
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|
6485 <pb n=105> |
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|
6486 <head>PROPOSITIO XXVIII.</head> |
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|
6487 <head>PROBLEMA.</head> |
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|
6488 <p>Propo$itum $it nobis efficere, potentiam pon |
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|
6489 dus mouentem, & pondus per data $patia $ibi in |
|
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|
6490 uicem longitudine commen$urabilia moueri. |
|
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|
6491 <p>Sit datum $patium potentiæ, vt tria, <marg><I>Ex</I> 22 <I>huius.</I></marg> |
|
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|
6492 ponderis verò, vt quatuor. inueniatur po |
|
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|
6493 tentia in A pondus B $u$tinens, quæ pon |
|
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|
6494 deris $it $e$quitertia, vt quatuor ad trìa. $i |
|
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|
6495 igitur in A $it potentia mouens pondus; <marg><I>Ex eadem.</I></marg> |
|
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|
6496 erit $patium ponderis $patii potentiæ $e$- |
|
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|
6497 quitertium, vt quatuor ad tria. quod face |
|
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|
6498 re oportebat. |
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6499 <fig> |
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6500 <p>Hoc autem & ex iis, quæ dicta $unt in |
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6501 vige$ima $ecunda, & in vige$imaquinta |
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6502 huius efficere po$$umus $olo fune. Quòd $i |
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6503 pluribus funibus id efficere voluerimus, |
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6504 non $olum multis, $ed infinitis modis hoc |
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6505 efficere poterimus, vt $upra dictum e$t. <marg><I>In</I> 26 <I>huius.</I></marg> |
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6506 Quare hoc affirmare po$$umus, quod qui- |
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6507 dem mirum e$$e videtur: videlicet. |
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6508 <foot>Dd</foot> |
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6509 <pb> |
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6510 <head>COROLLARIVM. I.</head> |
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6511 <p>Ex his manife$tum e$$e, Quamlibet datam in |
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6512 numeris proportionem inter pondus, & poten |
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6513 tiam; & inter $patium ponderis moti, & $patium |
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6514 potentiæ motæ; infinitis modis trochleis inueni- |
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6515 ri po$$e. |
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6516 <head>COROLLARIVM II.</head> |
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6517 <p>Ex dictis etiam manife$tum e$t, quò pondus |
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6518 facilius mouetur, eò quoq; tempus maius e$$e; |
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6519 quò verò difficilius, eò minus e$$e. & è con- |
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6520 uer$o. |
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6521 <pb n=106> |
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6522 <head>DE AXE IN |
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6523 PERITROCHIO.</head> |
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6524 <fig> |
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6525 <p>Fabricam, & cõ$tructionem hu- |
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6526 ius in$trumenti Pappus in octauo |
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6527 mathematicarum collectionum |
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6528 libro docet; axemq; vocat AB, |
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6529 tympanum verò CD circa idem |
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6530 centrum; & $cytalas in foramini- |
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6531 bus tympani EF GH & c. ita vt potentia, |
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6532 <foot>Dd 2</foot> |
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6533 <pb> |
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6534 <fig> |
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6535 quæ $emper in $cytalis e$t, vt in F, dum circum- |
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6536 uertit tympanum, & axem, $ur$um moueat pon- |
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6537 dus K axi appen$um fune LM circa axem reuo |
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6538 luto. Nobis igitur re$tat, vt o$tendamus, cur ma- |
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6539 gna pondera ab exigua virtute, quouè etiam mo |
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6540 do hoc in$trumento moueantur; temporis quin |
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6541 etiam, $patiiq; mouentis inuicem potentiæ, ac |
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6542 moti ponderis rationem aperiamus; huiu$modi- |
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6543 que in$trumenti v$um ad vectem reducamus. |
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6544 <pb n=107> |
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6545 <head>PROPOSITIO I.</head> |
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6546 <p>Potentia pondus $u$tinens axe in peritrochio |
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6547 ad pondus eandem habet proportionem, quam |
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6548 $emidiameter axis ad $emidiametrum tympani |
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6549 vná cum $cytala. |
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|
6550 <fig> |
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6551 <p>Sit diameter axis AB, cuius centrum C; $it diameter tympani |
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6552 DCE circa idem centrum; $intq; AB DE in eadem recta linea; |
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6553 $int deinde $cytalæ in foraminibus tympani DF GH & c.inter $e $e |
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6554 æquales, atq; æquè di$tantes; $itq; FE horizonti æquidi$tans; |
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6555 <pb> |
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6556 <fig> |
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6557 pondus autem K in fune BL circa axem volubili $it appen$um. & |
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6558 potentia in F $u$tineat pondus K. Dico potentiam in F ad pondus |
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6559 k ita $e habere, vt CB ad CF. fiat vt CF ad CB, ita pondus |
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6560 k ad aliud M, quod appendatur in F. & quoniam pondera M k |
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6561 appen$a $unt in FB; erit FB tanquam vectis, $iue libra; quia ve |
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6562 rò Ce$t punctum immobile, circa quod axis, tympanusq; reuol- |
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6563 uuntur; erit C fulcimentum vectis FB; vellibræ centrum. cùm |
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6564 <marg>6. <I>Primi Archim. de æquepon.</I></marg> autem it a $it CF ad CB, vt k ad M, pondera k M æqueponde- |
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6565 rabunt. Potentia igitur in F $u$tinens pondus k, ne deor$um ver- |
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6566 gat, ponderi K æqueponderabit; ip$iq; M æqualis erit. idem enim |
|
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6567 præ$tat potentia, quod pondus M. pondus igitur K ad poten |
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6568 <marg><I>Cor.</I> 4. <I>quinti.</I></marg> tiam in F erit, vt CF ad CB; & conuertendo, potentia ad |
|
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6569 pondus erit, vt CB ad CF, hoce$t, $emidiameter axis ad $emi |
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6570 <pb n=108> |
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6571 diametrum tympani vnà cum $cytala DF. Similiter etiam o$ten- |
|
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6572 detur, $i potentia pondus $u$tinens fuerit in Q. tunc enim $u$ti- |
|
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6573 neret vecte CQ; & ad pondus eam haberet proportionem, quam <marg>2 <I>Huuius. de vecte.</I></marg> |
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6574 habet CB ad CQ. Videlicet $emidiameter axis ad $emidiame- |
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6575 trum tympani vná cum $cytala EQ. quod demon$trare opor- |
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6576 tebat. |
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6577 <head>COROLLARIVM.</head> |
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6578 <p>Manife$tum e$t potentiam $emper minorem |
|
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6579 e$$e pondere. |
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6580 <p>Semidiameter enim axis $emper $emidiametro tympani mi- |
|
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|
6581 nor e$t. & potentia eò minor e$t pondere, quò $emidiameter axis |
|
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6582 minor e$t $emidiametro tympani vná cum $cytala. quare quò lon |
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6583 gior e$t CF, vel CQ; & quò breuior e$t CB, minor adhuc $em |
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6584 per potentia in F, vel in Q pondus k $u$tinebit. quò enim minor |
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6585 e$t CB, eò minorem habebit proportionem $emidiameter axis |
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6586 ad $emidiametrum tympani vná cum $cytala. |
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6587 <p>Hoc autem loco con$iderandum occurrit, quòd $i in alia $cyta- |
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6588 la appendatur pondus, vt in T, $u$tinens pondus k; it a nempè, vt |
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6589 pondus in T appen$um, pondusq; k circa axem con$titutum |
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6590 maneant; erit pondus in T grauius pondere M in F appen$o. |
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6591 iungatur enim TB, & à puncto C horizonti perpendicularis du- |
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6592 catur CI, quæ lineam TB $ecet in I; tandemq; connectatur |
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6593 TC, quæ æqualis erit CF. Quoniam autem pondera appen$a |
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6594 $unt in TB, perindè $e $e habebunt, ac $i in punctis TB ip$orum |
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6595 centra grauitatum haberent; vt antca dictum e$t. & quia ma- |
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6596 nent, erit punctum I (ex prima huius de libra) amborum $imul |
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6597 grauitatis centrum; cùm $it CI horizonti perpendicularis. $ed |
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6598 quoniam angulus BCI e$t rectus, erit BIC acutus, lineaq; BI <marg><I>Ex</I> 19 <I>primi.</I></marg> |
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6599 ip$a BC maior erit. quare angulus CIT erit obtu$us; atq; <marg><I>Ex</I> 13 <I>primi.</I></marg> |
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6600 ideo line^{a} CT ip$a T^{I} maior erit. Cùm autem CT maior $it |
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6601 TI, & IB maior BC; maiorem habebit proportionem TC ad |
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6602 CB, quàm TI ad IB; & conuertendo, minorem habebit pro- |
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6603 <pb> |
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6604 <fig> |
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6605 portionem BC ad CT, hoc e$t ad CF, quàm BI ad IT; vt ex |
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6606 vige$ima $exta quinti elementorum (iuxta Commandini editio- |
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6607 nem) patet. Quoniam verò punctum I e$t ponderum in TB |
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6608 <marg>6. <I>Primi Archim. de æquepon.</I></marg> exi$tentium centrum grauit atis; erit pondus in T ad pondus in B, |
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6609 vt BI ad IT. pondus verò in F ad idem pondus in Be$t, vt BC |
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6610 ad CF; maiorem igitur proportionem habebit pondus in T ad |
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6611 pondus in B, quàm pondus in F ad idem pondus in B. ergo |
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6612 <marg>10. <I>Quinti.</I></marg> grauius erit pondus in T, quàm pondus in F. |
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6613 <p>Si verò loco ponderis in T animata potentia $u$tinens pon- |
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6614 dus k con$tituatur; quæ ita degrauet $e, ac $i in centrum mundi |
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6615 tendere vellet; quemadmodum $uapte natura efficit pondus in T |
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6616 appen$um; erit hæc eadem ponderi in T appen$o æqualis; alio- |
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6617 quin non $u$tineret; quæ quidem ip$a potentia in F collocata ma |
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6618 <pb n=109> |
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6619 ior erit. $icuti enim $e $e habet pondus in T ad pondus in F, ita |
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6620 & potentia in T ad potentiam in F; cùm potentiæ $int ponderi- |
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6621 bus æquales. verùm $i vnaquæq; potentia $eor$um $umpta, tàm |
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6622 in T, quàm in F $u$tinens pondus $ecundũ circũferentiam THFN |
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6623 moueri $e vellet, veluti apprehen$a manu $cytala; tunc eademmet |
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6624 potentia, vel in F, vel in T con$tituta idem pondus k $u$tinere po |
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6625 terit; cùm $emper in cuiu$cunq; extremitate $cytalæ ponatur, ab |
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6626 eodem centro C æquidi$tans fuerit, ac $ecundum eandem circum |
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6627 ferentiam ab eodem centro æqualiter $emper di$tantem perpen$io |
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6628 nem habeat. neq; enim ($icuti pondus) proprio nutu magis in |
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6629 centrum ferri exoptat, qu<*>m circulariter moueri; cùm vtrunq;, $eu |
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6630 quemlibet alium motum nullo pror$us re$piciat di$crimine. pro- |
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6631 pterea non eodem modo res $e $e habet, $iue pondera, $iue anímatæ |
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6632 potentiæ ii$dem locis eodem munere abeundo fuerint con$titutæ. |
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6633 <p>Potentia autem mouet pondus vecte FB, videlicet dum po |
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6634 tentia in F circumuertit tympanum, circumuertit etiam axem; & |
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6635 FB fit tamquam vectis, cuius fulcimentum C, potentia mouens |
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6636 in F, & podus in B appen$um. & dum punctum F peruenit in N; |
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6637 punctum H erit in F, & punctum B erit in O; ita vt ducta NO |
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6638 tran$eat per C; eodemq; tempore pondus k motum erit in P, ita |
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6639 vt OBP $it æqualis ip$i BL, cùm $it idem funis. |
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6640 <p>Deinde ex quarta huius de vecte facilè eliciemus $patium po- |
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6641 tentiæ mouentis ad $patium ponderis moti ita e$$e, vt $emidiame |
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6642 ter tympani cùm $cytala ad $emidiametrum axis, hoc e$t, vt CF |
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6643 ad CB, cùm circumferentia FN ad BO, $it vt CF ad CB. & quo <marg><I>Ex</I> 4 <I>huius de vecte.</I></marg> |
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6644 niam BL, e$t æqualis OBP, dempta communi BP, erit OB ip |
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6645 $i PL æqualis. quare FN $patium potentiæ ad PL $patium pon- |
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6646 deris erit, vt CF ad CB, videlicet $emidiameter tympani cùm |
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6647 $cytala ad $emidiametrum axis. Quod idem o$tendetur, poten- |
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6648 tia vel in Q, vel in qualibet alia $cytala exi$tente, vt in S. cùm |
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6649 enim $cytalæ $int $ibi inuicem æquales, atq; æqualiter di$tantes; |
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6650 vbicunq; $it potentia æquali mota velocitate $emper æquali tem- |
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6651 pore æquale $patium pertran$ibit, hoc e$t ex Q in R, vel ex Sin T |
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6652 eodem tempore mouebitur, quò ex F in N. $ed quò tempore po |
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6653 tentia ex Fin N mouetur, eodemmet pror$us pondus k ex L in |
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6654 P quoq; mouetur; vbicunq; igitur $it potentia, erit $patium poten- |
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6655 <foot>Ee</foot> |
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6656 <pb> |
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6657 <fig> |
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6658 tiæ ad $patium ponderis moti, vt CF ad CB, hoc e$t $emidia- |
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6659 meter tympani cum $cytala, ad $emidiametrum axis. |
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6660 <head>COROLLARIVM. I.</head> |
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6661 <p>Ex his manife$tum e$t, ita e$$e pondus ad po- |
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6662 tentiam pondus $u$tinentem, vt $patium poten- |
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6663 tiæ mouentis ad $patium ponderis moti. |
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6664 <pb n=110> |
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6665 <head>COROLLARIVM II.</head> |
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6666 <p>Manife$tum e$t etiam, maiorem $emper ha- |
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6667 bere proportionem $patium potentiæ mouentis |
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6668 ad $patium ponderis moti, quàm pondus ad ean |
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6669 dem potentiam. |
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6670 <p>Præterea quò circulus FHN circa $cytalas e$t maior, eò quoq; |
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6671 in pondere mouendo maius $umetur tempus; dummodo potentia |
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6672 æquali moueatur velocitate. tempu$q; eò maius erit, quò diame |
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6673 ter vnius diametro alterius e$t maior. circulorum enim circumfe. <marg>23 <I>Octaui libri Pappi.</I></marg> |
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6674 rentiæ ita $e habent, vt diametri. Cùm vero ex trige$ima $exta |
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6675 quarti libri Pappi Mathematicarum collectionum, duorum inæ |
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6676 qualium circulorum æquales circumferentias inuenire po$simus; |
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6677 ideo tempus quoq; portionum circulorum inæqualium hoc modo |
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6678 inueniemus. è conuer$o autem, quò maior erit axis circumferen |
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6679 tia citius pondus $ur$um mouebitur. maior enim pars funis BL |
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6680 in vna circumuer$ione completa circa circulum ABO reuoluitur, |
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6681 quàm $i minor e$$et; cùm funis circumuolutus $it circumferen- |
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6682 tiæ circuli æqualis, circa quem reuoluitur. |
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6683 <head>COROLLAR VM.</head> |
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6684 <p>Ex his manife$tum e$t, quò facilius pondus mo |
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6685 uetur, tempus quoq; eò maius e$$e; & quò dif- |
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6686 ficilius, eò tempus minus e$$e. & è conuer$o. |
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6687 <foot>Ec 2</foot> |
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6688 <pb> |
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6689 <head>PROPOSITIO II.</head> |
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6690 <head>PROBLEMA.</head> |
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6691 <p>Datum pondus à data potentia axe in peritro- |
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6692 chio moueri. |
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6693 <p>Sit datum pondus $exagin |
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6694 ta; potentia verò vt decem. |
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6695 exponatur quædam recta li- |
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6696 nea AB, quæ diuidatur in C, |
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6697 ita vt AC ad CB eandem |
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6698 <fig> |
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6699 habeat proportionem, quam $exaginta ad decem. & $i CB axis |
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6700 $emidiameter e$$et, & CA $emidiameter tympani cùm $cytalis; |
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6701 <marg><I>Per præce dentem.</I></marg> patet potentiam vt decem in A ponderi $exaginta in B æquepon |
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6702 derare. Accipiatur autem inter BC quoduis punctum D; fiatq; |
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6703 BD $emidiameter axis, & DA $emidiameter tympani cùm $cy- |
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6704 talis; ponaturq; pondus $exaginta in B fune circa axem, & potentia |
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6705 <marg><I>Lemma in primi buius de vecte.</I></marg> <I>in A. Quoniam enim AD ad DB maiorem habet proportio- |
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6706 nem, quam AC ad CB; maiorem habebit proportionem AD ad |
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6707 DB, quam pondus $exaginta in B appen$um ad potentiam vt decem</I> |
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6708 <marg><I>Ex</I> 11 <I>huius de vecte.</I></marg> in A. Quare potentia in A pondus $exaginta axe imperitro- |
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6709 chio mouebit, cuius axis $emidiameter e$t BD, & DA $emidia |
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6710 meter tympani cùm $cytalis. quod erat faciendum. |
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|
6711 <pb n=111> |
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6712 <head>ALITER.</head> |
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6713 <head>Organicè verò melius erit hoc pacto.</head> |
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6714 <p>Exponatur axis, cuius |
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6715 diameter $it BD, & cen- |
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6716 trum C, quem quidem |
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6717 axem maiorem, vel mino |
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6718 rem con$tituemus, veluti |
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6719 <fig> |
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6720 magnitudo, ponderi$q; grauitas po$tulat. producatur deinde BD |
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6721 v$q; ad A: fiatq; BC ad CA, vt decem ad $exaginta. & $i CA tym |
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6722 pani cùm $cytalis $emidiameter e$$et, potentia decem in A ponde |
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6723 ri $exaginta in B æqueponderaret. producatur verò BA ex parte |
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6724 A, & in hac producta linea quoduis accipiatur punctum E; fiatq; |
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6725 CE $emidiameter tympani cùm $cytalis; ponaturq; potentia vt |
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6726 decem in E; habebit EC ad CB maiorem proportionem, quàm |
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6727 pondus $exaginta in B ad potentiam vt decem in E. potentia igi- |
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6728 tur vt decem in E mouebit pondus $exaginta in B appen$um fune |
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6729 circa axem, cuius $emidiameter e$t CB, & CE $emidiameter tym |
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6730 pani cùm $cytalis. quod facere oportebat. |
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6731 <pb> |
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6732 <p>Sub hoc facultatis genere $unt ergatæ, $uccu- |
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6733 læ, terebræ, tympana cum $uis axibus, $iue dentata, |
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6734 $iue non; & $imilia. |
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6735 <p>Terebra verò habet etiam ne$cioquid cochleæ; dum enim mo- |
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6736 uet pondus, $cilicet dum perforat, ex $ua ferè natura $emper vlte- |
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6737 rius progreditur<I>:</I> habet enim ferè helices tamquam circa conum |
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6738 de$criptas. quoniam autem verticem habet acutum, ad cunei quoq; |
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6739 rationem commodè referri poterit. |
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6740 <fig> |
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6741 <pb n=112> |
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6742 <head>DE CVNEO.</head> |
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6743 <p>Aristoteles in quæ$tioni- |
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6744 bus Mechanicis quæ$tione deci- |
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6745 ma$eptima a$$erit, cuneum $cin- |
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6746 dendo ponderi duorum vicem |
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6747 pror$us gerere vectium $ibi inui- |
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6748 cem contrariorum hoc niodo. |
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6749 <p>Sit cuneus ABC, cu |
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6750 ius vertex B, & $it AB |
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6751 æqualis BC; quod au |
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6752 tem $cindendum e$t, |
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6753 $it DEFG; $itq; pars |
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6754 cunei HB k intra DE |
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6755 FG, & HB æqualis |
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6756 $it ip$i Bk. percutiatur |
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6757 (vt fieri $olet) cuneus |
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6758 in AC, dum cuneus in |
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6759 AC percutitur, AB fit |
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6760 vectis, cuius fulcimen |
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6761 tum e$t H, & pondus in |
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6762 B. eodemq; modo CB |
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6763 fit vectis, cuius fulci- |
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|
6764 <fig> |
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6765 mentum e$t K, & pondus $imiliter in B. $ed dum percutitur cu- |
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6766 neus, maiori adhuc ip$ius portione ip$um DEFG ingreditur, |
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6767 quàm prius e$$et: $it autem portio hæc MBL; $itq; M Bip$i BL |
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6768 æqualis. & cùm MB BI. $int ip$is HB BK maiores; erit ML maior |
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|
6769 <pb> |
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6770 Hk. dum igitur ML |
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6771 erit in $itu Hk; opor- |
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|
6772 ter, vt fiatmaior $ci$sio; |
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|
6773 & D moueatur ver$us |
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6774 O, G autem ver$us N: |
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6775 & quò maior pars cu |
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|
6776 nei intra DEFG ingre |
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|
6777 dietur, eò maior fiet |
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6778 $ci$sio; & DG ma- |
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|
6779 gis adhuc impellentur |
|
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|
6780 ver$us ON. pars igi |
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6781 tur KG eius, quod $cin |
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|
6782 ditur, mouebitur à ve- |
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6783 cte AB, cuius fulcimen |
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6784 tum e$t H, & pondus |
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|
6785 <fig> |
|
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|
6786 in B; ita vt punctum B ip$ius vectis AB impellat partem KG. |
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6787 & pars HD mouebitur à vecte CB, cuius fulcimentum e$t k; ita |
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6788 vt B vecte CB partem HD impellat. |
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|
6789 <p>Cùm autem tria $int vectium genera, vt $upra |
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|
6790 o$ten$um e$t; idcirco conuenientius erit forta$$è |
|
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|
6791 cuneum hoc modo con$iderare. |
|
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|
6792 <p>Ii$dem po$itis, intelligatur vectis AB, cuius fulcimentum B, & |
|
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|
6793 pondus in H, vt in $ecunda huius de vecte diximus. $imiliter ve- |
|
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|
6794 ctis CB, cuius fulcimentum B, & pondus in K; ita vt pars HD |
|
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|
6795 moueatur à vecte AB, cuius fulcimentum e$t B, & pondus in H; |
|
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|
6796 ita vt punctum H ip$ius vectis AB impellat partem HD. $imi |
|
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|
6797 li quoq; modo pars KG moueatur à vecte CB, cuius fulcimentum |
|
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|
6798 e$t B, & pondus in k, it aut k ip$ius uectis CB partem k G mo- |
|
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|
6799 ueat. quod quidem for$itan rationi magis con$entaneum erit. |
|
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|
6800 <pb n=113> |
|
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|
6801 <p>Sit enim cuneus ABC; |
|
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|
6802 $intq; duo pondera $epa- |
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|
6803 rat a DEFG, & HIkL, |
|
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|
6804 intra quæ $it pars cunei |
|
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|
6805 DBH, cuius uertex B |
|
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|
6806 medium inter utrumq; $i |
|
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6807 tum obtineat. percutia- |
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6808 tur autem cuneus, ita ut |
|
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|
6809 magis adhuc intra pon- |
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|
6810 dera propellatur, $icuti |
|
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|
6811 prius dictum e$t; ponde- |
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|
6812 <fig> |
|
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|
6813 ra enim $unt, ac $i unum tantùm continuum e$$et GFkL, quod |
|
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|
6814 $cindendum e$$et: eodem enim modo pars DG, dum cuneus |
|
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|
6815 ulterius impellitur, mouebitur uer$us M; & pars HL uer$us N. |
|
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|
6816 Moueatur itaq; pars DG uer$us M, & pars HL uer$us N, B uerò |
|
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|
6817 dum ulterius progreditur, $emper medium inter utrunq; pondus |
|
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|
6818 remaneat. dum autem DG à cuneo mouetur uer$us M; patet B |
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|
6819 non mouere partem DG uer$us Muecte CB, cuius fulcimentum |
|
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6820 H; punctũ enim B non tangit pondus; $ed DG mouebitur à pun- |
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|
6821 cto uectis Duecte AB, cuius fulcimentum B; punctum enim D tan |
|
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|
6822 git pondus, & in$trumenta mouent per contactum. Similiter |
|
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|
6823 HL mouebitur ab H uecte CB, cuius fulcimentum B; & uterq; |
|
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|
6824 uectis utriq; re$i$tit in B, ita ut B potius fulcimenti uice fungatur, |
|
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|
6825 quàm mouendi ponderis. quod ip$um hoc quoq; modo manife- |
|
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|
6826 $tum erit. |
|
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6827 <foot>Ff</foot> |
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6828 <pb> |
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6829 <p>Sit, quod $cindendum e$t A |
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6830 BCD parallelogrammũ rectan- |
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6831 gulum; $intq; duo vectes æqua- |
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6832 les EF GF, & partes vectium |
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6833 HF KF $int intra ABCD; $itq; |
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6834 HF æqualis Fk, & HA æqua |
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6835 lis KB. Oporteat verò vecti- |
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6836 bus EF GF $cindere ABCD |
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6837 ab$q; percu$sione, videlicet $int |
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6838 potentiæ mouentes in EG æqua |
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6839 les. vt autem $cindatur ABCD, |
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6840 oportet partem HA moueriuer |
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6841 <fig> |
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6842 $us M. & kB ver$us N; $ed dum vectes mouentur, putá alter in |
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6843 M, alter verò in N; nece$$e e$t, vt punctum F immobile rema |
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6844 neat; in illo enim fit vectium occur$us. quare F erit fulcimen- |
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6845 tum vtriu$q; vectis, & FG mouebit partem kB, cuius fulcimen |
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6846 tum erit F, & potentia mouens in G; & pondus in k. $imi- |
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6847 liter pars HA mouebitur à vecte EF, cuius fulcimentum F, po |
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6848 tentia in E, & pondus in H. |
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6849 <p>Si autem k H e$$ent fulcimenta immobilia, & pondera in F; |
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6850 dum vectis FG conatur mouere pondus in F, tunc ei re$i$tit ve- |
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6851 ctis EF, qui etiam conatur mouere pondus in F ad partem op |
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6852 po$itam; $ed quoniam potentiæ $unt æquales, & cætera æqualia; |
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6853 ergo in Fnon fiet motus: æquale enim non mouet æquale. patet |
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6854 igitur in F maximam fieri vectium $ibi inuicem occurrentium re$i |
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6855 $tentiam, itaut F $it quoddam immobile. Quare con$iderando |
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6856 cuneum, vtmouet vectibus $ibi inuicem aduer$is, for$itan eis po |
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6857 tius utitur hoc $ecundo modo, quàm primo. |
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6858 <p>Quoniam autem totus cuneus $cindendo mo |
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6859 uetur, po$$umus idcirco eundem alio quoq; mo |
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6860 do con$iderare; videlicet dum ingreditur id, |
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6861 <pb n=114> |
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6862 quod $cinditur, nihil aliud e$$e, ni$i pondus $u |
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6863 pra planum horizonti inclinatum mouere. |
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6864 <fig> |
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6865 <p>Sit planum horizonti æquidi$tans tran$iens per AB; $it cuneus |
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6866 CDB, & CD æqualis ip$i DB; & latus cunei DB $it $emper in |
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6867 $ubiecto plano. $it deinde pondus AEFG immobile in A; $itq; |
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6868 pars cunei EDH $ub AEFG. Quoniam enim dum percutitur cu |
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6869 neus in CB, maior pars cunei ingreditur $ub AEFG, quàm $it |
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6870 EDH; $it hæc pars IDH. & quoniam latus cunei DB $emper |
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6871 e$t in $ubiecto plano per AB ducto horizonti parallelo, tunc quan |
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|
6872 do pars cunei kDI erit $ub AEFG; erit punctum k in H, & I |
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6873 $ub E. $ed Ik maior e$t HE; punctum igitur E $ur$um motum |
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6874 erit. & dum cuneus $ub AEFG ingreditur, punctum E $ur$um |
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6875 $uper latus cunei EI mouebitur, eodemq; modo $i cuneus vlterius |
|
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|
6876 progredietur, $emper punctum E $uper latus cunei DC mouebitur: |
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6877 punctum igitur E ponderis $uper planum DC mouebitur horizonti |
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6878 inclinatum, cuius inclinatio e$t angulus BDC. quod demon- |
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|
6879 $trare oportebat. |
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|
6880 <foot>Ff 2</foot> |
|
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6881 <pb> |
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6882 <fig> |
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6883 <p>In hoc exemplo, con$iderando cuneum in$tar vectis mouen- |
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6884 tem, manife$tum e$t, cuneum BCD pondus AEFG vecte CD |
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6885 mouere; ita vt D $it fulcimentum, & pondus in E. non autem ve |
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6886 cte BD, cuius fulcimentum H, & pondus in D. |
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6887 <p>Vt autem res clarior reddatur, alio vtamur |
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|
6888 exemplo. |
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6889 <p>Sit planum hori- |
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|
6890 zonti æquidi$tans |
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6891 tran$iens per AB; $it |
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6892 cuneus CAB, cuius |
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6893 latus AB $it $emper |
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6894 in $ubiecto plano; $it- |
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|
6895 qué pondus AEFG, |
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6896 quod nullum alium |
|
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6897 habeat motum, ni$i |
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6898 <fig> |
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6899 $ur$um, & deor$um ad rectos angulos horizonti; ita vt ducta IGk |
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6900 $ubiecto plano, ip$iqué AB perpendicularis, punctum G $it $em |
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6901 per in linea IGk. & quoniam dum cuneus percutitur in CB, to |
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6902 tus $uper AB vlterius progreditur; pondus AEFG eleuabitur ex |
|
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6903 <pb n=115> |
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6904 iis, quæ $upra diximus. Moueatur cuneus ita, vt E tandem per- |
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6905 ueniat in C, & po$itio cunei ABC $it MNO, & po$itio pon- |
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6906 deris AEFG $it PMQI, & G$it in I. Quoniam itaq; dum cu |
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6907 neus $uper lineam BO mouetur, pondus AEFG $ur$um moue- |
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6908 tur à linea AC. & dum cuneus ABC vlterius progreditur, $em |
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6909 per pondus AEFG magis à latere cunei AC eleuatur: pondus igi |
|
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6910 tur AEFG $uper planum cunei AC mouebitur; quod quidem |
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6911 nihil aliud e$t, ni$i planum horizonti inclinatum, cuius inclinatio |
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6912 e$t angulus BAC. |
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6913 <p>Hic motus facilè ad libram, vectemq; reducitur. quod enim |
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6914 $uper planum horizonti inclinatum mouetur ex nona Pappi octa- |
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6915 ui libri Mathematicarum collectionum reducitur ad libram. ea- |
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6916 dem enim e$t ratio, $iue manente cuneo, vt pondus $uper cunei |
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6917 latus moueatur; $iue eodem etiam moto, pondus adhuc $uper ip |
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6918 $ius latus moueatur; tamquam $uper planum horizonti incli- |
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6919 natum. |
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6920 <p>Ea verò, quæ $cinduntur, quomodo tam- |
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6921 quam $uper plana horizonti inclinata mouean. |
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6922 tur, o$tendamus. |
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6923 <p>Sit cuneus ABC, |
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6924 & AB ip$i BC æqua- |
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6925 lis. Diuidatur AC |
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6926 bifariam in D, conne- |
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6927 ctaturq; BD. $it dein- |
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|
6928 de linea EF, per quam |
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|
6929 tran$eat planum hori |
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6930 zonti æquidi$tans; $itq; |
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6931 BD in eadem linea EF; |
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6932 & dum cuneus percuti |
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6933 tur, dumq; mouetur ver |
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|
6934 <fig> |
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6935 $us E, $emper BD $it in linea EF. quod verò $cindendum e$t |
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6936 $it GHLM, intra quod $it pars cunei kBI. manife$tum e$t, |
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|
6937 <pb> |
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|
6938 dum cuneus uer$us E |
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6939 mouetur, partem kG |
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6940 ver$us N moueri; & par |
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6941 tem HI uer$us O. per |
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6942 cutiatur cuneus, ita vt |
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6943 AC $it in linea NO; |
|
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6944 tunc k erit in A, & I in |
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6945 C: & k ex $uperius di |
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|
6946 ctis motum erit $uper |
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6947 kA, & I $uper IC. |
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6948 quare dum cuneus mo |
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|
6949 <fig> |
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6950 uetur, pars KG $uper BA latus cunei mouebitur, & pars IH $uper |
|
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6951 latus BC. pars igitur kG $uper planum mouetur horizonti incli- |
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6952 natum, cuius inclinatio e$t angulus FBA. $imiliter IH moue- |
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6953 tur $uper planum BC in angulo FBC. Partes ergo eius, quod |
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6954 $cinditur $uper plana horizonti inclinata mouebuntur. & quam- |
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6955 quam planum BC $it $ub horizonte; pars tamen IH $uper IC mo |
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6956 uetur, tamquam $i BC e$$et $upra horizont&etilde; in angulo DBC. partes |
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6957 enim eius quod $inditur, eodem tempore, ab cadem potentia mo- |
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6958 uentur; eadem ergo erit ratio motus partis IH, ac partis KG. $i- |
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6959 militer eadem e$t ratio, $iue EF $it horizonti æquidi$tans, $iue |
|
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|
6960 horizonti perpendicularis, vel alio modo. nece$$e e$t enim poten |
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|
6961 tiam cuneum mouentem eandem e$$e, cùm cætera eadem rema |
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|
6962 neant. eadem igitur erit ratio. |
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6963 <p>Po$t hæc con$iderandum e$t, quæ nam $int ea, quæ efficiunt, |
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6964 vt aliquod facilius moueatur, $iue $cindatur. quæ quidem duo |
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|
6965 $unt. |
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6966 <p>Primum, quod efficit, vt aliquod facilè $cin |
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6967 datur, quod etiam ad e$$entiam cunei magis per- |
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6968 tinet, e$t angulus ad verticem cunei; quò enim |
|
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6969 minor e$t angulus, eò facilius mouet, ac $cindit. |
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|
6970 <pb n=116> |
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6971 <p>Sint duo cunei ABC DEF, & angulus |
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6972 ABC ad verticem minor $it angulo DEF. |
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|
6973 dico aliquod facilius moueri, $iue $cindi à cu |
|
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|
6974 neo ABC, quàm à DEF. diuidantur AC |
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|
6975 DF bifariam in G H punctis; connectan- |
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|
6976 turq; BG, & EH. Quoniam enim partes |
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6977 eius, quod $cinditur à cuneo ABC, $u- |
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|
6978 per planum horizonti inclinatum mouen- |
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|
6979 tur, cuius inclinatio e$t GBA: quæ ve- |
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|
6980 rò à cuneo DEF, $uper planum horizonti |
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|
6981 inclinatum mouentur, cuius inclinatio e$t |
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|
6982 <fig> |
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6983 HED; & angulus GBA minor e$t angulo HED; cùm |
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|
6984 CBA minor $it DEF: & ex nona Pappi octaui libri mathe |
|
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6985 maticarum collectionum, quod mouetur $uper planum AB faci- |
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6986 lius mouebitur, & à minore potentia, quàm $uper ED; Quod |
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6987 ergo $cinditur à cuneo ABC facilius, & à minore potentia $cin |
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6988 detur, quàm à cuneo DEF. $imiliter o$tendetur, quò magis an- |
|
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6989 gulus ad verticem cunei erit acutus, eò facilius aliquod moueri, |
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|
6990 ac $cindi. quod demon$trare oportebat. |
|
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|
6991 <p>Po$$umus etiam hoc alia ratione o$tendere |
|
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|
6992 con$iderando cuneum, vt vectibus $ibi inuicem |
|
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|
6993 aduer$is mouet, $icuti $ecundo modo dictum e$t. |
|
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6994 hoc autem prius o$tendere oportet. |
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6995 <pb> |
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6996 <p>Sit vectis AB, cuius fulcimentum |
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6997 $it B immobile; quod autem mouen- |
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6998 dum e$t, $it CDEF rectangulum ita |
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6999 accommodatum, vt deor$um ex par |
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|
7000 te FE mouerinon po$sit; & punctum |
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7001 E$it immobile, & tamquam centrum; |
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7002 ita vt punctum D moueatur per cir- |
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7003 cumferentiam circuli DH, cuius cen- |
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7004 trum $it E. & C per circumferentiam |
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7005 CL, ita vt iuncta CE $it eius $emi |
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7006 diameter. tangat in$uper CDEF ve |
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|
7007 <fig> |
|
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7008 ctem AB in C, atq; vectis AB moueat pondus CDEF, & po |
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7009 tentia mouens $it in A, fulcimentum B, & pondus in C. $it |
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7010 deinde alius vectis MCN, qui etiam moueat CDEF, cuius ful |
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7011 cimentum immobile $it N; potentia mouens in M, & pondus |
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7012 $imiliter in C; $itq; CN æqualis ip$i CB, & CM ip$i CA; al |
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7013 ternatimq; moueatur pondus CDEF vectibus AB MN. dico |
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7014 CDEF facilius ab eadem potentia moueri vecte AB, quàm ve |
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7015 cte MN. |
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7016 <p>Fiat centrum B, & interuallo BC circumferentia de$cribatur |
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7017 CO. $imiliter centro N, interuallo quidem NC, circumferen |
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7018 tia de$cribatur CP. Quoniam enim dum vectis AB mouet CD |
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7019 EF, punctum vetis C mouetur $uper circumferentiam CO; cùm |
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7020 $it B fulcimentum, & centrum immobile. $imiliter dum vectis |
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7021 MN mouet CDEF, punctum C mouetur per circumferentiam |
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7022 CP; dum igitur vectis AB mouet CDEF, conatur mouere pun |
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7023 ctum C ponderis $uper circumferentiam CO; quod quidem effi |
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7024 cere non pote$t: quia C mouetur$uper circumferentiam CL. qua |
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7025 re in motu vectis AB $ecundùm partem ip$i re$pondentem, ac mo |
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7026 tu ponderis $ecundum C facto, contingit repugnantia quædam; |
|
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7027 in diuer$as enim partes mouentur. $imiliter dum vectis MN mo |
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7028 uet CDEF, conatur mouere C $uper circumferentiam CP; at- |
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|
7029 que ideo in hoc etiam vtroq; motu $imilis oritur repugnantia. |
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7030 quoniam autem circumferentia CO propior e$t circumferentiæ |
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7031 CL, quam $it CP; hoc e$t propior e$t motui, quem facit pun- |
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7032 ctum C ponderis; ideo minor erit repugnantia inter motum vectis |
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|
7033 <pb n=117> |
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7034 AB, & motum C ponderis, quàm inter motum vec tis MN, & |
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|
7035 motum eiu$dem C. quod etiam patet, $i intelligatur CF hori- |
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7036 zonti perpendicularis, tunc enim circumferentia CP magis ten |
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|
7037 dit deor$um, quàm CO; & CL tendit $ur$um. & ideo minor fit re |
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|
7038 pugnantia inter vectem AB, & motum C, quàm inter vect&etilde; MN, & |
|
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|
7039 motum C. $ed vbi minor repugnantia ibi maior facilitas. ergo faci |
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|
7040 liusmouebitur CD EF vecte AB, quàm vecte MN. quod demon |
|
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|
7041 $trare oportebat. |
|
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|
7042 <head>COROLLARIVM.</head> |
|
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|
7043 <p>Ex hoc manife$tum e$t, quò minor e$t an- |
|
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|
7044 gulus à linea CF, vel CE, vel CD contentus; |
|
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|
7045 hoc e$t, quò minor e$t angulus BCF, vel BCE, |
|
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|
7046 vel etiam BCD, eò facilius pondus moueri. |
|
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|
7047 quod quidem eodem modo o$tendetur. |
|
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|
7048 <p>Quod autem propo$itum e$t, $ic demon- |
|
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|
7049 $trabimus. |
|
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|
7050 <p>Sint cunei ABC DE |
|
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|
7051 F, & angulus ABC mi- |
|
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|
7052 nor $it angulo DEF, & |
|
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|
7053 AB BC DE EF $int in |
|
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|
7054 ter $e $e æquales. Sint de- |
|
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|
7055 inde quatuor pondera æ- |
|
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|
7056 qualia GH IL NO QR |
|
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|
7057 rectan gula; $intq; LM |
|
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|
7058 kH in eadem recta linea: |
|
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|
7059 <fig> |
|
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|
7060 $imiliter RS PO in recta linea; erunt GK IM parallelæ, & NP <marg><I>Ex</I> 28 <I>primi.</I></marg> |
|
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|
7061 QS parallelæ. $it IBG pars cunei intra pondera GH IL; & cu |
|
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|
7062 nei pars QEN intra pondera NO QR; $intqué IB BG QE |
|
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|
7063 EN inter $e $e æquales. dico pondera GH IL facilius ab eadem |
|
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|
7064 <foot>Gg</foot> |
|
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|
7065 <pb> |
|
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|
7066 potentia moueri cuneo |
|
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|
7067 ABC, quàm pondera |
|
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|
7068 NO QR cuneo DEF. |
|
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|
7069 <p>Diuidantur AC DF |
|
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|
7070 bifariam in TV, iungan |
|
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|
7071 turq; TBVE, erunt an- |
|
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|
7072 guli ad T, & V recti. con |
|
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|
7073 nectatur IG, quæ $ecet |
|
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|
7074 BT in X. Quoniam e- |
|
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|
7075 <fig> |
|
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|
7076 nim IB e$t æqualis BG, & BA æqualis BC; erit IA ip$i GC |
|
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|
7077 <marg>2 <I>Sexti.</I></marg> æqualis. quare vt BI ad IA, ita e$t BG ad GC. parallela igitur |
|
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7078 <marg><I>Ex</I> 29 <I>primi.</I></marg> e$t IG ip$i AC. ac propterea anguli ad X $unt recti: $ed & an |
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7079 <marg>28 <I>Primi.</I></marg> guli XG k XIM $unt recti, rectangulum enim e$t GM; quare |
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7080 TB æquidi$tans e$t ip$is Gk IM. angulus igitur TBC æqua- |
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7081 lis e$t angulo BGK, & TBA ip$i BIM æqualis. $imiliter demon |
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7082 $trabimus angulum VEF æqualem e$$e ENP, & VED æqualem |
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7083 EQS. cùm autem angulus ABC minor $it angulo DEF; erit |
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7084 & angulus TBC minor VEN. quare & BGk minor ENP. |
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7085 $imili modo BIM minor EQS. quoniam autem cuneus ABC |
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7086 duobus mouet vectibus AB BC, quorum fulcimenta $unt in B; |
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7087 & pondera in GI: $imiliter cuneus DEF duobus vectibus mouet |
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7088 DE EF, quorum fulcimenta $unt in E; & pondera in N Q: per |
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7089 præcedentem pondera GH IL facilius vectibus AB BC mo- |
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7090 uebuntur, quàm pondera NO QR vectibus DE EF. ponde- |
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7091 ra ergo GH IL facilius cuneo ABC mouebuntur, quàm ponde- |
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7092 ra NO QR cuneo DEF. & quia eadem e$t ratio in mouendo, |
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7093 atq, in $cindendo; facilius idcirco aliquod cuneo ABC $cindetur |
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7094 quàm cuneo DEF. $imiliterq; o$tendetur, quò minor e$t angu |
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7095 lus ad verticem cunei, eò facilius aliquod moueri, vel $cindi. quod |
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7096 demon$trare oportebat. |
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7097 <p>Præterea quæ mouentur à cuneo DEF, per maiora mouentur |
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7098 $patia; quàm ea, quæ à cuneo ABC. nam vt DF $it intra QN, |
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7099 & AC $it intra IG; nece$$e e$t, vt QN per $patia moueantur |
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7100 maiora; $cilicet vnum dextror$um, alter $ini$tror$um, quàm IG; |
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7101 cùm DF maior $it AC; dummodo totus cuneus intra pondera in- |
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7102 <pb n=118> |
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7103 grediatur. à potentia verò facilius eodem tempore mouetur ali- |
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7104 quod per minus $patium, quàm per maius; dummodo cætera, qui- |
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7105 bus fit motus, $int æqualia: $i ergo eodem tempore AC DF in |
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7106 IG QN perueniãt, cùm AI CG DQ FN $int inter$e $e æqua |
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7107 les; facilius à potentia mouebuntur GI cuneo ABC, quàm QN |
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7108 cuneo DEF. quare facilius pondera GH IL à potentia mouebun |
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7109 tur cuneo ABC, quàm pondera NO QR cuneo DEF. $imiliter- |
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7110 qué o$tendetur, quò angulus ad verticem cunei minor e$$et, eò fa |
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7111 cilius pondera moueri, vel $cindi. |
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7112 <p>Secundum, quod efficit, vt aliquod facilius |
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7113 $cindatur, e$t percu$sio; qua cuneus mouetur, & |
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7114 mouet; hoc e$t percutitur, ac $cindit. |
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7115 <fig> |
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7116 <p>Sit cuneus A, quod $cinditur B, quod |
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7117 percutit C; quod quidem, vel ex $e ip$o, |
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7118 vel à regente, atq; ip$um mouente poten |
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7119 tia percutit, atq; mouet. $i quidem ex |
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7120 $e ip$o, Primùm quò grauius erit, eò |
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7121 maior fiet percu$sio. quinetiam, quò |
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7122 longior fuerit di$tantia inter AC, maior |
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7123 itidem fiet percu$sio. graue enim vnum- |
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7124 quodq; dum mouetur; grauitatis ma- |
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7125 gis a$$umit motum, quàm quie$cens: & |
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7126 adhuc magis quo longius mouetur. |
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7127 <fig> |
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7128 <foot>Gg 2</foot> |
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7129 <pb> |
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7130 <p>Si verò C ab aliqua moueatur po |
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7131 tentia, vt $i per manubrium DE mo |
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7132 ueatur; primùm quò grauius erit C, |
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7133 deinde quò longius erit DE, eò ma- |
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7134 ior fiet percu$sio. $i enim ponatur po |
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7135 tentia mouens in E, erit C magis di |
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7136 $tans à centro & ideo citius mouebi |
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7137 tur. vt in quæ$tionibus Mechanicis |
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7138 latè mon$trat Ari$toteles; nec non |
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7139 ex iis, quæ in tractatu de libra di- |
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7140 cta fuere, patere pote$t, quò magis |
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7141 <fig> |
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7142 pondus Cà centro di$tat, eò grauius reddi. quod ip$um etiam va |
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7143 lidiori pellet impul$u virtute in E potentiore exi$tente. |
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7144 <p>Hoc verò $ecundùm e$t, quod efficit, vt hoc in$trumento ma- |
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7145 gna moueantur, $cindanturq; pondera. percu$sio enim vis e$t ua |
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7146 lidi$sima, vt ex decimanona quæ$tionū Mechanicarum Ari$totelis |
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7147 patet. $i enim $upra cuneum maximum imponatur onus; tunc cu- |
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7148 neus nihil ferè efficiet, præ$ertim ictus comparatione. quod $i ad |
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7149 huc ip$i cuneo vectem, vel cochleam, vel quoduis aliud huiu$mo |
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7150 di aptetur in$trumentum ad cuneum ponderi intimius propellen- |
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7151 dum, nullius ferè momenti præ ictu continget effectus. cuius qui- |
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7152 <pb n=119> |
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7153 dem rei inditio e$$e pote$t, $i fuerit |
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7154 corpus A lapideũ, ex quo aliquam eius |
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7155 partem detrahere qui$piam voluerit, pu |
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7156 tá partem anguli B; tunc malleo ferreo |
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7157 ab$q; alio in$trumento percutiendo in B, |
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7158 facilè aliquam anguli B partem franget. |
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7159 quod quidem nullo alio in$trumento |
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7160 percu$sionis munere carente, ni$i maxi |
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7161 ma cùm difficultate efficere poterit; $iue |
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7162 <fig> |
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7163 fuerit vectis, $iue cochlea, $iue quoduis aliud huiu$modi. quare |
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7164 percu$sio in cau$a e$t, quo magna $cindantur pondera. cùm autem |
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7165 $ola percu$sio tantam vim habeat, $i ei aliquod adiiciamus in$tru |
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7166 mentum ad mouendum, $cindendumq; accomodatum, admiran |
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7167 da profectò videbimus. In$trumentum huiu$ |
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7168 modi cuneus e$t, in quo duo (quantum ad ip- |
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7169 $ius formam attinet) con$ideranda occurrunt. |
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7170 Alterum e$t, cuneum ad $u$cipiendam, $u$tinen |
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7171 damq; percu$sionem apti$simum e$$e; alterum |
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7172 e$t quòd propter eius in altera parte $ubtilita- |
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7173 tem facilè intra corpora ingreditur, vt manife |
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7174 $tè patet. Cuneus ergo cum percu$sione ip$ius |
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7175 efficit, vt in mouendis, $cindendi$q; ponderi- |
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7176 bus ferè miracula cernamus. |
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7177 <fig> |
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7178 <pb> |
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7179 <p>Ad huiu$modi facultatis in$trumentum, ea |
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7180 quoquè omnia commodè referri po$$unt, quæ |
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7181 percu$sione, $iue impul$u incidunt, diuidunt, |
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7182 perforant, huiu$modiq; alia obeunt munera. vt |
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7183 en$es, gladii, mucrones, $ecures, & $imilia. $erra |
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7184 quoq; ad hoc reducetur; dentes enim percu- |
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7185 tiunt, cuneiq; in$tar exi$tunt. |
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7186 <pb n=120> |
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7187 <head>DE COCHLEA.</head> |
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7188 <p>Pappvs in eodem octauo libro |
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7189 multa pertractans de cochlea, do |
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7190 cet quomodo conficienda $it; & |
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7191 quomodo magna huiu$modi in- |
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7192 $trumento moueantnr pondera; |
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7193 nec non alia theoremata ad eius |
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7194 cognitionem valdè vtilia. Quoniam autem in- |
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7195 ter cætera pollicetur, $e o$tendere velle, co- |
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7196 chleam nihil aliud e$$e præter a$$umptum cu- |
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7197 neum per cu$sionis expertem vecte motionem |
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7198 facientem; hoc autem in ip$o de$ideratur; pro- |
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7199 pterea idip$um o$tendere conabimur, nec non |
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7200 eiu$dem cochleæ ad vectem, libramq; reductio- |
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7201 nem; vt ip$ius tandem completa habeatur co- |
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7202 gnitio. |
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7203 <pb> |
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7204 <fig> |
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7205 <p>Sit cuneus ABC, qui circa cylindrum DE circumuoluatur:$itq; |
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7206 IGH cuneus circa cylindrum reuolutus, cuius vertex $it I. $it de- |
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7207 inde cylindrus cum circumpo$ito cuneo ita accomodatus, vt ab$q; |
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7208 vllo impedim&etilde;to manubrio kF eius axi annexo circumuerti po$sit. |
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7209 $itq; LMNO, quod $cindendum e$t; quod etiam ex parte MN |
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7210 $it immobile: vt in iis, quæ $cinduntur, fieri $olet: & $it vertex |
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7211 I intra RS. circumuertatur kF, & perueniat ad kP; dum autem kF |
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7212 circumuertitur, circumuertitur etiam totus cylindrus DE, & cu- |
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7213 neus IGH: quare dum KF erit in kP, vertex I non erit amplius |
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7214 intra RS, $ed cuneipars alia, vt TV: $ed TV maior e$t, quàm |
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7215 RS; $emper enim pars cunei, quæ magis à vertice di$tat, maior |
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7216 e$tea, quæ ip$i e$t propinquior: vt igitur TV $it intra RS, opor- |
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7217 tet, vt R cedat, moueaturq; ver$us X, & S ver$us Z, vt faciunt |
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7218 ea, quæ $cinduntur. totum ergo LMNO $cindetur. $imiliter |
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7219 què demon$trabimus, dum manubrium kP erit in kQ, tunc GH |
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7220 e$$e intra RS: & vt GH $it intra RS, nece$$e e$t, vt R $it in X, |
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7221 & S in Z; ita vt <I>X</I>Z $it æqualis GH; $emperq; LMNO amplius |
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7222 $cindetur. $icigitur patet, dum kF circumuertitur, $emper R moue |
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7223 ri ver$us X, atq; S ver$us Z: & R $emper $uper ITG moueri, S au |
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7224 tem$uper IVH, hoc e$t $uper latera cunei circa cylindrum circum |
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7225 uoluti. |
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|
7226 <pb n=121> |
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7227 <head>PROPOSIO I.</head> |
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7228 <p>Cuneus hoc modocirca cylindrum accommo- |
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7229 datus, nihil e$t aliud; ni$i cochlea duas habenshe |
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7230 lices in vnic o punctoinuicem coniunctas. |
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7231 <fig> |
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7232 <p>Sit cuneus ABC; & AB |
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7233 ip$i BC æqualis. diuidatur |
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7234 AC bifariam in D, iunga |
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7235 turq; BD; erit BD ip$i AC |
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7236 perpendicularis; & AD |
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7237 ip$i DC æqualis, triangu- |
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7238 lumq; ABD triangulo C |
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7239 BD æquale. fiant deinde |
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7240 triangula rectangula EFG |
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7241 HIk non $olum inter $e, |
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7242 verùm etiam vtriq; ADB |
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7243 & CDB æqualia. $itq; cy |
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7244 lindrus LMNO, cuius perimeter $it æqualis vtriq; FG kI. & |
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7245 LMNO $it parallelogrammum per axem. fiatq; MP æqualis |
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7246 FE; & PN æqualis HI. ponaturq; HI in NP, circumuolua- |
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7247 turq; triangulum HIk circa cylindrum; & $ecundùm kH helix |
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7248 de$cribatur NQP, vt Pappus quoq; docet in octauo libro propo |
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7249 $itione vige$ima quarta. $imiliter ponatur EF in MP, circum- |
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7250 uoluaturq; triangulum EFG circa cylindrum; de$cribaturq; per |
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7251 EG helix PRM. cùm itaq; PMPN $int æquales EFHI, erit |
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7252 MN æqualis ip$i AC, & cùm helices PRM PQN $int æquales |
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7253 lineis EGHk; helices igiturip$is ABBC æquales erunt. cu- |
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7254 neus ergo ABC totus circumuolutus erit circa cylindrum LMNO. |
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7255 <foot>Hh</foot> |
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7256 <pb> |
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7257 incidantur deinde helices, |
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7258 vt docet Pappus $ecundùm |
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7259 latitudinem cunei; & hoc |
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7260 modo cuneus vná cum cy |
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7261 lindro nihil aliud erit, |
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7262 quàm cochlea duas habens |
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7263 helices PRMPQN cir |
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7264 ca cylindrum LN in vnico |
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7265 puncto P inuicem coniun |
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7266 ctas. quod demon$trare o- |
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7267 portebat. |
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7268 <fig> |
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7269 <head>COROLLARIVM.</head> |
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7270 <p>Hinc manife$tum e$$e pote$t, quomodo heli- |
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7271 ces in ip$a cochlea de$cribi po$sint. |
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7272 <p>Quomodo autem pondera $uper helices co- |
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7273 chleæ moueantur, o$tendamus. |
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7274 <fig> |
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7275 <p>Sit (veluti prius) cuneus IGH circa cylindrum DE reuolutus, |
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7276 cuius vertex $it I. apteturq; cylindrus ita, vt liberè vna cum $uo |
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7277 axe circumuertatur. $intq; duo pondera MN cuiu$cunq; figuræ |
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7278 voluerimus, ita tamen aptata, vt moueri non po$sint, ni$i $uper |
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7279 <pb n=122> |
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7280 rectam lineam LO, quæ axi cylindri $it æquidi$tans. $intq; MN |
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7281 iuxta cunei verticem I. Circumuertatur KF, & perueniat ad kP: |
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7282 dum autem kF erit in kP, tunc TV erit intra pondera MN; $i- |
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7283 cut $upra diximus. M igitur ver$us L mouebitur, & N ver$us O. |
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7284 $imiliter o$tendetur, dum kP erit in KQ, tunc GH e$$e intra pon- |
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7285 dera MN; & M erit in X, & N in Z; ita vt XZ $it æqualis GH. |
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7286 quare dum kF circumuertitur, $emper pondus N mouetur ver$us |
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7287 O, & $uper helicem IRS; M verò $uper aliam helicem. |
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7288 <fig> |
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7289 <p>Similiter$i cochlea plures habeat hæ- |
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7290 lices, vt in $ecunda figura, pondus A, |
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7291 dum cochlea circumuertitur, $emper $u- |
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7292 per helices BCDEFG mouebitur; |
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7293 dummodo pondus A aptetur ita vt mo- |
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7294 uerinon po$sit, ni$i $uper rectam HI ip$i |
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7295 cylindro æquidi$tantem. eodem enim |
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7296 modo, quo $uper primam mouetur heli |
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7297 cem, mouetur etiam $upra $ecundam, |
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7298 & tertiam, & cætera. quotcunq; enim |
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7299 fuerint helices, nihil aliud $unt, quàm |
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7300 latus cunei circa idem cylindrum iterum |
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7301 atq; iterum circumuolutum. & $iue co- |
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7302 chlea fuerit horizonti perpendicularis, |
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7303 $iue horizonti æquidi$tans, vel alio mo- |
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7304 do collocata, nihil refert: $emper enim |
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7305 cadem erit ratio. |
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7306 <foot>Hh 2</foot> |
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7307 <pb> |
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7308 <fig> |
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7309 <p>Si verò (vt in tertia figura) $upra cochleam imponatur aliquod, |
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7310 vt B, quod quidem tylum vocant, ita accommodatum, vt inferio |
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7311 ri parte helices habeat concauas ip$i cochleæ appo$itè admodum |
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7312 congruentes; per$picuum $atis e$$e poterit, ip$um B, dum coclhea |
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7313 circumuertitur, $uper helices cochleæ eo pror$us modo moueri; |
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7314 quo pondus iuxta primam figurã mouebatur: dummodo tylum ap- |
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7315 tetur, vt docet Pappus in octauo libro; ita $cilicet vt tantùm an- |
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7316 tè, retrouè axi cylindri æquidi$tans moueatur. |
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7317 <fig> |
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7318 <p>Et $i loco tyli, quod helices habet concauas in parte inferiori, con |
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7319 $tituatur, vt in quarta figura, cylindrus concauus vt D, & in eius |
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7320 concaua $uperficie de$cribantur helices, in cidanturq; ita, vt aptè |
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7321 <pb n=123> |
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7322 cùm cochlea congruant (eodem enim modo de$cribentur helices |
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7323 in $uper$icie concatia cylindri, $icuti fit in conuexa) $i deiade co- |
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7324 chlea in $uis polis firmetur, $cilicet in $uo axe, cir cumuertaturq;; |
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7325 patet D ad motum circumuer$ionis cochleæ quemmadmodum ty |
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7326 lum moueri. nec non $i D in EF firmetur, ita vt immobilis ma |
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7327 neat, dum circumuertitur cochlea; $uper helices cylindri D, ad |
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7328 motum $uæ circumuer$ionis dextror$um, vel $ini$tror$um factæ; |
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7329 tùm in anteriorem, tùm in po$teriorem partem mouebitur. cylin- |
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7330 drus autem D hoc modo accõmodatus vulgò mater, $iue cochleæ |
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7331 fæmina nuncupatur. |
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7332 <fig> |
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7333 <p>Si autem cochleæ (vt in quinta figura) tympanum C dentibus |
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7334 obliquis dentatum apponatur, vt docet Pappus in eodem octauo li- |
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7335 bro; vel etiam rectis; ita tamen con$tructis, vt facilècum cochlea |
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7336 conueniant: $imiliter manife$tum e$t ad motum cochleæ circumuer |
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7337 ti etiam tympanum C. eodemq; modo tympani dentes $uper he |
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7338 lices cochleæ moueri. & hæc dicitur cochlea infinita, quia & co |
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7339 chlea, & tympanum dum circumuertuntur, $emper eodem modo |
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7340 $e $e habent. |
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7341 <pb> |
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7342 <p>Hæc diximus, vt manife$tum $it cochleam in mouendo pondere |
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7343 cunei munere ab$q; percu$sione fungi. Illud enim remouet à loco, |
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7344 vbi erat; quemadmodum cuneus remouetea, quæ mouet, ac $cindit. |
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7345 omnia enim hæc à cochlea mouentur, $icuti pondus A in $ecun- |
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7346 da figura, & M in prima. |
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7347 <p>Quoniam autem duplici ratione mouentem cuneum con$iderari |
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7348 po$$e o$tendimus, videlicet vt mouet vectibus, vel vt e$t planum |
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7349 horizonti inclinatum, dupliciter quoq; cochleam con$iderabimus; |
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7350 <fig> |
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7351 & primùm vt vectibus mouet, vt in prima figura circumuertatur |
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7352 kF, & perueniat in KP; tunc, $icut dictum e$t, TV erit intra pon- |
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7353 dera MN. & $icut con$ideramus vectes in cuneo, eodem quoq; |
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7354 modo eos con$iderare po$$umus in cochlea hoc pacto. erit $cilicet |
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7355 IVH vectis, cuius fulcimentum I, & pondus in V. $imiliter ITG ve |
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7356 ctis, cuius fulcimentum I, & pondus in T. potentiæ verò mo- |
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7357 uentes GH e$$e deberent; $ed $icuti in cuneo potentia mouens |
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7358 e$t percu$sio, quæ mouet cuneum; idcirco erit, ubi potentia mo- |
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7359 uet cochleam; $cilicet in P manubrio kP. cochlea enim $ine per- |
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7360 cu$sione mouetur. Hæc autem con$ideratio propter vectes infle- |
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7361 xos impropria for$itan e$$e videbitur; Quocirca $i id, quod moue |
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7362 tur à cochlea, $upra planum horizonti inclinatum moueri intelli |
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7363 gatur; erit quidem huiu$modi con$ideratio (cùm ip$i quoq; cuneo |
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7364 conueniat) figuræ ip$ius cochleæ magis conformis. |
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|
7365 <pb n=124> |
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|
7366 <head>PROPOSITIO II.</head> |
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7367 <p>Si fuerit cochlea AB helices habens æquales |
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7368 CDEFG. Dico has nihil aliud e$$e præter pla |
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7369 num horizonti inclinatum circa cylindrum re- |
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|
7370 uolutum. |
|
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|
7371 <fig> |
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|
7372 <p>Sit cochlea AB horizonti perpendicularis duas habens helices |
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7373 CDEFG. exponatur HI æqualis GC, quæ bifariam diui- |
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|
7374 datur in k; erunt Hk kI non $olum inter $e $e, verùm etiam |
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|
7375 ip$is GE EC æquales, & ip$i HI ad rectos angulos ducatur LI; |
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7376 & per LI intelligatur planum horizonti æquidi$tans; $itq; LI du |
|
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7377 pla perimetro cylindri AB, quæ bifariam diuidatur in M; erunt |
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|
7378 IM ML cylindri perimetro æquales. connectatur HL, & à pun |
|
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|
7379 cto M ducatur MN ip$i HI æquidi$tans, coniungaturq; KN. quo |
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|
7380 niam enim $imilia $unt inter $e $e triangula HILNML, cùm <marg><I>Ex</I> 4. <I>$exti.</I></marg> |
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|
7381 <pb> |
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|
7382 <fig> |
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7383 NM $it æquidi$tans HI; erit LI ad IH, vt LM ad MN: & |
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7384 permutando vt IL ad LM; ita HI ad NM. $ed IL dupla e$t ip$ius |
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7385 LM; ergo & HI dupla erit MN. $ed e$t etiam dupla ip$ius kI, |
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|
7386 quare kI NM inter $e æquales erunt. & quoniam anguli ad MI |
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7387 $unt recti; erit kM parallelogrammum rectangulum, & kN æqua |
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7388 lis erit IM. quare KN perimetro cylindri AB æqualis erit. pona |
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7389 tur itaq; HI in GC, erit Hk in GE. circumuoluatur deinde trian |
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7390 gulum HkN circa cylindrum AB, de$cribet HN helicen GFE; |
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7391 cùm NK perimetro cylindri $it æqualis; & punctum N erit in E; |
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7392 & MN in CE. & quia ML æqualis e$t perimetro cylindri; cir- |
|
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7393 cumuoluatur rur$us triangulum NML circa cylindrum AB, NL |
|
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|
7394 de$cribet helicen EDC. quare tota LH duas de$cribet helices |
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7395 CDEFG. patet igitur has helices cochleæ nihil aliud e$$e, ni- |
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7396 $i planum horizonti inclinatum; cuius inclinatio e$t angulus HLI |
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7397 circa cylindrum circumuolutum, $upra quod pondus mouctur. |
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7398 quod demon$trare oportebat. |
|
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7399 <p>Quomodo autem hoc ad libram reducatur mnnife$tum e$t ex |
|
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|
7400 nona octaui libri eiu$dem Pappi. |
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|
7401 <pb n=125> |
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|
7402 <p>Po$tquam vidimus quomodo pondera huiu$modi moueantur |
|
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7403 in$trumento; nunc con$iderandum e$t, quæ nam $int ea, quæ effi |
|
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7404 ciunt, vt pondera facilè moueantur: hæc autem duo $unt. |
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7405 <p>Primùm quidem, quod efficit, vt facilè pon- |
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7406 dus moueatur, quod etiam ad e$$entiam cochleæ |
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7407 magis pertinere videtur; e$t helix circa co- |
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7408 chleam. vt $i circa datam cochleam AB duæ |
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7409 $int helices inæquales CDA EFG, $itq; AC mi |
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|
7410 nor EG. Dico idem pondus facilius $uper heli |
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7411 cen CDA moueri, quàm $uper EFG. |
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|
7412 <fig> |
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7413 <p>Compleatur cuneus |
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7414 ADCHI, hoc e$t de- |
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7415 $cribatur helix CHI |
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7416 æqualis CDA, & ver- |
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7417 tex cunei $it C. $imili |
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|
7418 ter compleatur cuneus |
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7419 GFEKL, cuius ver- |
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|
7420 tex E. exponatur de- |
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|
7421 inderecta linea MN, |
|
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7422 quæ $it ip$i AC æqua- |
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|
7423 lis, cui ad rectos angu |
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|
7424 los ducatur NP, quæ $it |
|
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|
7425 æqualis perimetro cy- |
|
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|
7426 lindri AB: & conne- <marg>1 <I>Huius.</I></marg> |
|
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|
7427 ctatur PM; erit PM, |
|
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|
7428 per ea, quæ dicta $unt, |
|
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|
7429 ip$i CDA æqualis. |
|
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|
7430 producatur deinde M |
|
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|
7431 N in O, fiatq; ON æ- |
|
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|
7432 qualis MN, coniunga |
|
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|
7433 turq; OP; erit OPM cuneus cuneo ADCHI æqualis. $imili- <marg>1 <I>Huius.</I></marg> |
|
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|
7434 <foot>Ii</foot> |
|
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|
7435 <pb> |
|
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|
7436 terq; exponatur cu- |
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|
7437 neus STQ æqualis cu |
|
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|
7438 neo GFEkL; erit TR |
|
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|
7439 ip$i PN, & perime- |
|
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|
7440 tro cylindri æqualis; & |
|
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|
7441 QR æqualis GE. |
|
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|
7442 cùm autem GE ma- |
|
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|
7443 ior $it AC; erit & RQ |
|
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|
7444 maior MN. $ecetur |
|
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|
7445 RQ in V; fiatq; RV |
|
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|
7446 ip$i MN æqualis, & |
|
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|
7447 coniungatur TV; erit |
|
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|
7448 triangulum TVR tri- |
|
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|
7449 angulo MPN æquale: |
|
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|
7450 duæ enim TR RV |
|
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|
7451 duabus PN NM $unt |
|
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|
7452 æquales, & anguli, |
|
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|
7453 quos continent, $unt |
|
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|
7454 æquales, nemperecti; |
|
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|
7455 <marg>4 <I>Primi.</I></marg> angulus igitur RTV |
|
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|
7456 <fig> |
|
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|
7457 angulo NPM æqualis erit. quare angulus MPN minor e$t angu- |
|
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|
7458 lo QTR; & horum dupli, angulus $cilicet MPO minor angulo |
|
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|
7459 QTS. quoniam autem cuneus, qui angulum ad verticem mino |
|
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|
7460 rem habet, facilius mouet, ac $cindit, quàm qui habet maiorem; |
|
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|
7461 cuneus ergo MPO facilius mouebit, quàm QTS. facilius igitur |
|
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|
7462 pondus à cuneo ADCHI mouebitur, quàm à cuneo GFEkL. |
|
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|
7463 pondus ergo $uper helicen CDA facilius mouebitur, quàm $uper |
|
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|
7464 EFG. eodemq; modo o$tendetur, quò minor erit AC, eò faci- |
|
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|
7465 lius pondus moueri. quod demon$trare oportebat. |
|
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|
7466 <pb n=126> |
|
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|
7467 <fig> |
|
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|
7468 <head>ALITER.</head> |
|
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|
7469 <p>Sit data cochlea AB duas habens helices æquales CDEFG; $it |
|
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|
7470 deinde alius cylindrus <G>ab</G> ip$i AB æqualis, in quo $ummatur OP ip |
|
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|
7471 $i CG æqualis; diuidaturq; OP in tres partes æquales OR RT |
|
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|
7472 TP, & tres de$cribantur helices OQRSTVP; erit vnaquæq; OR RT |
|
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|
7473 TP minor CE, & EG: tertia enim pars minor e$t dimidia. dico |
|
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|
7474 idem pondus facilius $uper helices OQRSTVP moueri, quàm $u |
|
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|
7475 per CDEFG. exponatur HIL triangulum orthogonium, ita vt |
|
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|
7476 HI $it ip$i CG æqualis, & IL duplo perimetri cylindri AB æqua |
|
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|
7477 lis, & per <I>L</I>I intelligatur planum horizonti æqui$tans; erit H<I>L</I> |
|
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|
7478 æqualis CDEFG; & H<I>L</I>I inclinationis angulus erit. exponatur <marg><I>Ex</I> 2 <I>huius.</I></marg> |
|
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|
7479 $imiliter <I>X</I>YZ triangulum orthogonium, ita vt XZ ip$i OP $it æ- |
|
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|
7480 qualis, quæ etiam æqualis erit CG, & HI; $itq; ZY cylindri pe- |
|
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|
7481 rimetro tripla, erit XY æqualis OQRSTVP. diuidatur ZY in |
|
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|
7482 <foot>Ii 2</foot> |
|
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|
7483 <pb> |
|
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|
7484 <fig> |
|
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|
7485 tres partes æquales in <G>g</G><35>; erit vnàquæq; Z <G>g g</G> <35> <35> Y perimetro cy |
|
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|
7486 lindri <G>ab</G> æqualis, quæ etiã perimetro cylindri AB æquales erunt; & |
|
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|
7487 per con$equens ip$is IM, & ML. connectatur X<35>. & quoniam |
|
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|
7488 duæ HI IL duabus XZ Z<35> $unt æquales, & angulus HIL re- |
|
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|
7489 ctus æqualis e$t angulo XZ<35> recto; erit triangulum HIL trian- |
|
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7490 gulo XZ<35> æquale; & angulus HLI angulo X<35>Z æqualis; & |
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7491 <marg>21 <I>Primi.</I></marg> X<35> ip$i HL æqualis. $ed quoniam angulus X<35>Z maior e$t angu |
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7492 lo <I>X</I>YZ; erit angulus HLI angulo <I>X</I>YZ maior. ac propterea planũ |
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7493 HL magis horizonti inclinat, quàm XY. quare id&etilde; põdus à minore |
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7494 potentia $uper planũ XY, quàm $uper planũ HL mouebitur; vt faci |
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7495 lè elicitur ex ead&etilde; nona Pappi. cùm aut&etilde; helices OQRSTVP nihil |
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7496 aliud $int, quàm planũ XY horizonti inclinatũ in angulo XYZ cir |
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7497 ca cylindrum <G>ab</G> circumuolutum; & helices CDEFG nihil $unt |
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7498 aliud, quàm planum HL horizonti inclinatum in angulo HLI cir |
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7499 ca cylindrum AB circumuolutum; facilius ergo pondus $uper he- |
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7500 <pb n=127> |
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7501 lices OQRSTVP mouebitur, quàm $uper helices CDEFG. |
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7502 <p>Si autem OP diuidatur in quatuor partes æquales, de$cribantur- |
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7503 què circa <G>ab</G> quatuor helices; adhuc facilius pondus mouebitur $u- |
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7504 per has quatuor, quàm $uper tres OQRSTVP. & quò plures |
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7505 erunt helices, eò facilius pondus mouebitur. quod demon$trare |
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7506 oportebat. |
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7507 <p>Tempus verò huius motus facilè patet, helices enim CDEFG |
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7508 $unt æquales HL; helices verò OQRSTVP $unt æquales |
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7509 XY: $ed XY maior e$t HL; ideo fiat Y<G>e</G> ip$i HL æqualis: $i igi <marg><I>Ex</I> 18 <I>Primi.</I></marg> |
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7510 tur duo pondera $uper lineas LHY<I>X</I> moueantur, & veloci- |
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7511 tates motuum $int æquales, citius pertran$ibit quod mouetur $uper |
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7512 LH, quàm quod $uper Y<I>X</I> mouetur. in eodem enim tempore erunt |
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7513 in H<G>e</G>. quare tempus eius, quod mouetur $uper helices OQRS |
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7514 TVP, maius erit eo, quod e$t men$ura eius, quod mouetur $uper C |
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7515 DEFG. & quò plures erunt helices, eò maius erit tempus. cùm au |
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7516 tem datæ $int lineæ HI<I>XZ</I>, & IL<I>Z</I>Y: datæ enim $unt cochleæ AB |
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7517 <G>ab</G>; & anguli ad IZ recti dati; crit HL data. $imiliter & <I>X</I>Y data <marg><I>Ex</I> 48 <I>primi.</I></marg> |
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7518 erit. quare & harum proportio data erit. temporum igitur propor <marg>1 <I>Datorum & Ex $exta primi Ioannis de Monte rego de triangulis.</I></marg> |
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7519 tio eorum, quæ $uper helices mouentur data erit. |
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7520 <p>Alterum, quod efficit, vt pondera facilè mo- |
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7521 ueantur, $unt $cytalæ, aut manubria, quibus co- |
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7522 chlea circumuertitur. |
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7523 <pb> |
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7524 <fig> |
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7525 <p>Sit cochlea habens helices ABCD, quæ etiam $cytalas ha- |
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7526 beat EFGH foraminibus cochleæ impo$itas. $it infra helices |
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7527 cylindrus MN, in quo non $int inci$æ helices; & circa cylindrum |
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7528 funis circumuoluatur trahens pondus O, quod ad motum $cytala |
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7529 rum EFGH moueatur, ac $i ergatæ in$trumento traheretur. du |
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7530 catur (per ea quæ prius dicta $unt de axe in peritrochio) Lk $cy |
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7531 talæ æqualis, axiq; cylindri perpendicularis, eumq; $ecans in I: |
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7532 patet quò longior $it LI, & quò breuior $it Ik, pondus O facilius |
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7533 moueri. e$t autem animaduertendum, quòd dum cochlea mouet |
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7534 pondus, $i mente concipiatur, quòd loco trahendi pondus O fune, |
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7535 pondus $uper helices ABCD moueat; pondus quoq; in k, quod |
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7536 $it R, $uper helices etiam facilius mouebit. e$t enim LK vectis, cuius |
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7537 <marg>2 <I>Cor.</I></marg> fulcimentum e$t I: cùm circa axem cochlea circumuertatur; po- |
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7538 <marg>1 <I>huius de vecte.</I></marg> tentia mouens in L; & pondus in k. facilius enim mouetur pon |
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7539 dus vecte Lk, quàm $ine vecte; quia LI $emper maior e$t Ik. |
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7540 <pb n=128> |
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7541 Intelligatur itaq; manente cochlea pondus R moueri à potentia |
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7542 in L vecte Lk $uper helicen Ck: vel quod idem e$t, $icut etiam |
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7543 $upra diximus, $i pondus R aptetur ita, vt moueri non po$sit, ni |
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7544 $i $uper rectam PQ axi cylindri æquidi$tantem; circumuertaturq; |
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7545 cochlea, potentia exi$tente in L; mouebitur pondus R $uper he- |
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7546 licen CD eodem modo, ac $i à vecte Lk moueretur. idem enim |
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7547 e$t, $iue pondus manente cochlea $uper helicen moueatur; $iue he |
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7548 lix circumuertatur, ita vt pondus $uper ip$am moueatur. cùm |
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7549 ab eadem potentia in L moueatur. $imiliter o$tendetur, quò lon. |
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7550 gior $it LI, adhuc pondus facilius $emper moueri. à minori enim <marg><I>Ex</I> 1 <I>huius de vecte.</I></marg> |
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7551 potentia moueretur. quod erat propo$itum. |
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7552 <p>Tempus quoq; huius motus manife$tum e$t, quò enim longior |
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7553 e$t LI, eò tempus maius erit: dummodo potentiæ motuum $int |
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7554 in yelocitate æquales; $icuti dictum e$t de axe in peritrochio. |
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7555 <head>COROLLARIVM.</head> |
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7556 <p>Ex his manife$tum e$t. quò plures $unt heli- |
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7557 ces; & quò longiores $unt $eytalæ, $iue manu- |
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7558 bria, pondus ip$um facilius quidem, tardius au |
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7559 tem moueri. |
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7560 <p>Virtus deniq; mouentis, atq; in $cytalis con- |
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7561 $titutæ potentiæ, hinc manife$ta fiet. |
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7562 <pb> |
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7563 <fig> |
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7564 <p>Sit datum A centum; $it planum horizonti inclinatum CD in |
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7565 angulo DCE. inueniatur ex eadem nona Pappi quanta vi pondus |
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7566 A $uper CD mouetur; quæ $it decem. exponatur cochlea LM |
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7567 helices habens GHIK &c. in angulo ECD; per ea, quæ dicta |
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7568 $unt, potentia decem pondus A $uper helices GHIk mouebit. $i |
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7569 autem hac cochlea volumus pondus A mouere, & potentia mo- |
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7570 uens $it vt duo. ducatur NP axi cochleæ perpendicularis, axem |
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7571 $ecans in O; fiatq; PO ad ON, vt vnum ad quinq; hoc e$t duo ad |
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7572 <marg><I>Ex</I> 1 <I>huius de vecte.</I></marg> decem. Quoniam enim potentia mouens pondus A in P, ide$t |
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7573 $uper helices e$t vt decem, cui potentiæ re$i$tit, & æqualis e$t po |
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7574 tentia in N vt duo; e$t enim NP vectis, cuius fulcimentum e$t |
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7575 O. potentia ergo vt duo in N pondus A $uper helices cochleæ |
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7576 mouebit. efficiantur igitur $cytalæ, $iue manubria, quæ v$q; ad N |
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7577 <pb n=129> |
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7578 perueniant; manife$tum e$t, potentiam vt duo in his pondus cen- |
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7579 tum cochlea <I>L</I>M mouere. |
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7580 <p>Si igitur $it cochlea QR helices habens in angulo DCE, & cir- |
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7581 ca ip$am $it eius mater S, quæ $i pependerit centum, adiiciatur ST |
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7582 manubrium quoddam, $iue $cytala; ita vt T in eadem proportio- |
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7583 ne di$tet ab axe cylindri, vt NOP; patet potentiam vt duo in T |
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7584 mouere S $uper helices cochleæ. nihil enim aliud e$t S, ni$i pon- |
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7585 dus $uper helices cochleæ motum. $imiliter $i S $it immobilis, cir- |
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7586 cumuertaturq; cochlea manubrio, $iue $cytala QX in eadem pro- |
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7587 portione con$ecta; fueritq; cochlea centum pondo (quòd qui- |
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7588 dem, vel ex $e ip$a, vel cum pondere V cochleæ appen$o, vel cum |
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7589 pondere Y cochleæ $uper impo$ito centum pependerit) manife- |
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7590 $tum e$t potentiam vt duo in X mouere co chleam QR $uper he |
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7591 lices intra matricem cochleæ inci$as. atq; ita in aliis, quæ cochleæ |
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7592 in$trumento mouentur; proportionem potentiæ ad pondus inue- |
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7593 niemus. |
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7594 <head>COROLLARIVM.</head> |
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7595 <p>Ex hoc manife$tum e$t, quomodo datum pon |
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7596 dus à data potentia cochlea moueatur. |
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7597 <foot>Kk</foot> |
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7598 <pb> |
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7599 <fig> |
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7600 <p>Illud quoq; præterea hoc loco ob$eruandum occurrit; quò plu- |
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7601 res erunt matricis cochleæ helices, eò minus in pondere mouen- |
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7602 do cochleam pati. $i enim matrix vnicam duntaxat helicen po$$e |
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7603 derit, tunc pondus vt centrum à $ola cochleæ $u$tinebitur helice; |
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7604 $i verò plures, in plures quoque, ac totidem cochleæ heli- |
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7605 ces ponderis grauitas di$tribuetur; vt $i quatuor contineat helices, |
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7606 tunc quatuor vici$sim cochleæ helices vniuer$o ponderi $u$tin endo |
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7607 incumbent; $iquidem vnaquæquè quartam totius ponderis portio- |
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7608 nem $u$tentabit. quòd $i adhuc plures contineat helices, ponderis |
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7609 quoq; totius in plures, atque ideo minores portiones fiet di$tri- |
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7610 butio. |
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7611 <pb n=130> |
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7612 <p>O$ten$um e$t igitur pondus à cochlea moueri |
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7613 tamquam à cuneo percu$sionis experte: loco e- |
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7614 nim percu$sionis mouet vecte, hoc e$t $cytala, $i- |
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7615 ue manubrio. |
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7616 <p>His demon$tratis liquet, quomodo datũ pon- |
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7617 dus à data potentia moueri po$sit. quòd $i vecte |
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7618 hoc a$$equi volumus; po$$umus & dato vecte da |
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7619 tum pondus data potentia mouere. quod quidem |
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7620 in nullis ex aliis fieri po$$e ab$olutè contin git: $iue |
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7621 $it cochlea, $iue axis in peritrochio, $iue trochlea. |
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7622 non enim datis trochleis, neq; dato axe in peri- |
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7623 trochio, neq; data cochlea, datum pondus à data |
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7624 potentia moueri pote$t, cùm potentia in his $em- |
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7625 per $it determinata: $i igitur pot&etilde;tia, quæ pondus |
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7626 mouere debeat, hac minor $it data, nun quam pon |
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7627 dus mouebit. po$$umus tamen dato axe, & tympa- |
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7628 no ab$q; $cytalis datum pondus data pot&etilde;tia mo- |
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7629 uere; cùm $cytalas con$truere po$simus, ita vt $e |
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7630 midiameter tympani dati vná cum longitudine |
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7631 $cytalæ ad axis $emidiametrum datã habeat pro- |
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7632 portionem. quod idem cochleæ contingere po |
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7633 te$t, $cilicet datum pondus data cochlea $ine ma |
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7634 nubrio, vel $cytala, data potentia mouere. co- |
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7635 gnita enim potentia, quæ pondus $uper helices |
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7636 moueat, po$$umus manubrium, $iue $cytalam ita |
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7637 <foot>Kk 2</foot> |
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7638 <pb> |
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7639 con$truere, vt data potentia in $cytala eandem |
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7640 vim haheat, quam potentia pondus $uper helices |
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7641 mouens cùm autem hoc datis trochleis nullo mo |
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7642 do fieri po$sit. datum tamen pondus data poten- |
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7643 tia trochleis infinitis modis mouere po$$umus. |
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7644 datum verò pondus data potentia cunei in$tru- |
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7645 mento mouere, hoc minimè fieri po$$e clarum e$ |
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7646 $e videtur; non enim data potentia datum pon- |
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7647 dus $uper planum horizonti inclinatum mouere |
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7648 pote$t, neq; datum pondus à data potentia moue |
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7649 bitur vectibus $ibi inuic&etilde; aduer$is, quemmadmo- |
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7650 dum in cuneo in$unt; cùm in vectibus cunei pro- |
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7651 pria, veraq; vectis proportio $eruari non po$sit. |
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7652 vectium enim fulcimenta non $unt immobilia, |
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7653 cùm totus cuneus moueatur. |
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7654 <p>Poterit deinde quis $truere machinas, atq; eas |
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7655 ex pluribus componere; vt ex trochleis, & $uc- |
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7656 culis, vel ergatis, pluribu$uè dentatis tympanis, |
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7657 uel quocunq; alio modo; & ex ijs, quæ diximus; fa |
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7658 cilè inter pondus, & potentiam proportionem |
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7659 inuenire. |
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7660 <head>FINIS.</head> |
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7661 <pb> |
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7662 <head>Locorum aliquot, quæ inter imprimendum deprauata |
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7663 $unt, emendatior lectio.</head> |
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7664 <p><I>Pagina</I> 2, <I>b, ver$u</I> 19, <I>AEBD</I> ¶ 5, <I>a</I>, 6, <I>ip$i</I> ¶ 7, <I>b</I>, 9, <I>ODH</I> ¶ 9, <I>b</I>, 19, <I>cõting it</I> |
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7665 ¶ 15, <I>a</I>, 24, <I>grauius</I> ¶ 16, <I>b</I>, 30, <I>recto</I> ¶ 21, <I>a</I>, 26, <I>$u$tineatur</I> ¶ 23, <I>b</I>, 8, <I>BD DC</I> ¶ 31, <I>b</I>, |
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7666 9, <I>totum GK</I> ¶ 34, <I>a</I>, 24, <I>pondera FG</I> ¶ 38, <I>b</I>, 27, <I>maior AF</I> ¶ 39, <I>b</I>, 24 <I>AB in D</I> ¶ 40, |
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7667 <I>a</I>, 1, <I>ad BD</I> ¶ 44, <I>b</I>, 24, <I>graui</I> ¶ 48, <I>a</I>, 7, <I>ip$i AD</I> ¶ 50, <I>b</I>, 12 <I>pondus</I> ¶ 54, <I>a</I>, 7, <I>quàm</I> ¶ 61, |
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7668 <I>a</I>, 6, <I>præterquam in E</I> ¶ 65, <I>a</I>, 33, <I>quam</I> ¶ 81, <I>a</I>, 1, <I>ligato</I> ¶ 85, <I>b</I>, 22, <I>vtriq;</I> ¶ 97, <I>a</I>, 14, |
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7669 <I>dextror$um</I> ¶ 98, <I>b</I>, 20, <I>Hic</I> ¶ 110, <I>b, in po$till. Lemma in primã</I> ¶ 122, <I>a</I>, 8, <I>&</I> 17, <I>helicen</I> |
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7670 ¶ 123, <I>b</I>, 15, <I>ventes in GH</I> ¶ 124, <I>b</I>, 17, <I>manife$tum</I> ¶ 127, <I>a, in po$til. Monteregio</I> |
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7671 ¶ 127, <I>b, in po$til. ex Cor.</I> |
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7672 <head>REGISTRVM.</head> |
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7673 <head><12><12><12> ABCDEFGHIKLMNOPQRSTVX |
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7674 YZ, Aa Bb Cc Dd Ee Ff Gg Hh Ii Kk.</head> |
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7675 <head>Omnes duerni.</head> |
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7676 <head>PISAVRI</head> |
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7677 <head>Apud Hieronymum Concordiam.</head> |
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7678 <head>M. D. LXXVII.</head> |