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annotate texts/archimedesOldCVSRepository/archimedes/raw/cevag_geome_01_la_1692.raw @ 10:d7b79f6537bb

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Version vom 2009-02-14
author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Thu, 02 May 2013 11:08:12 +0200
parents 22d6a63640c6
children
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1 <pb>
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2 <FIG>
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3 <pb>
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4 <C>GEOMETRIA</C>
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5 <C>MOTUS</C>
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6 <C>OPVSCVLVM GEOMETRICVM</C>
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7 <C>A'</C>
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8 <C><I>IOANNE CEVA MEDIOLANENSI</I></C>
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9 <C>In gratiam Aquarum excogitatum.</C>
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10 <C>CONTINET DVOS LIBROS</C>
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11 <C>Primum de Simplici Motu,
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12 Alterum de Compo$ito.</C>
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13 <FIG>
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14 <C>BONONI&AElig;, M. DC. XCII.</C>
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15 <HR>
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16 <C>Typis HH. Antonij Pi$arij Superiorum permi$$u.</C>
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17 <pb>
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18 <C>SERENISSIMO</C>
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19 <C>MANTV&AElig; DUCI</C>
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20 <C>FERDINANDO
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21 CAROLO.</C>
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22 <p><I>ITerum, Sereni$$ime Princeps, tuis aduolutus
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23 genibus opu$culum exhibeo, in quo naturam motuum, pleniori
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24 methodo, qu&agrave;m puto antea $it actum, geometric&egrave; exequor.
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25 Nece&szlig;e habui h&aelig;c pr&aelig;mittere, qu&ograve; viam aperirem, &amp; quo-
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26 dammodo alueum $ternerem aquarum doctrin&aelig;, quarum
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27 argumentum vtili$$imum, &amp; profund&aelig; indaginis iam diu
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28 meditor. Quam arduum $it, &amp; per quas $alebras eun-
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29 dum, vt nouum aliquid luce dignum &egrave; latebris natur&aelig; eruarur
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30 vtinam Cel$itudini tu&aelig; aliquis veritatum non vulgarium
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31 indagator fidem faceret; $cio equidem, &amp; laboris improbitas
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32 tangeret benigni$$imum animum tuum, &amp; $imal natur&aelig; inge-
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33 nium $u$piceres, qu&aelig; mentibus aliquorum vim inuentricem
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34 in$eruit, vt eorum iugi cogitatione humanis v$ibus prouide-</I>
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35 <pb>
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36 <I>ret. Et ver&ograve; ($i in hoc genere de me quidquam confiteri decet)
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37 ni$i aduer$&aelig; valetudinis experimento prudentior factus indo-
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38 lem meam huiu$cemodi $tudijs intemperanter addictam ali-
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39 quot ab hinc annis compe$cui&szlig;em; nec non quotidie munus &agrave;
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40 Cel$itudine Tua $ummo cum honore &amp; beneficentia demanda-
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41 tum (adeo vt hoc etiam nomine Te$eruatorem meum appella-
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42 re po$$im) inde me reuoca$$et; eorum, credo equidem, ponderi,
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43 a$$idu&aelig;que contemplationi $uccumbere nece$$e erat. Vnde au-
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44 tem, Cel$i$$ime dux, huic $cienti&aelig; tanta vis, vt quos $ibi $emet
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45 adiunxerit, nonni$i altiori ratione queat a $e ip$a dimittere?
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46 An quod forta$$e vbi animus public&aelig; vtilitati de$eruire c&aelig;pe-
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47 rit, veluti in natar&aelig; concilium admi$$us, $ui quodammodo
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48 oblitus, propriam humilioremque $edem reui$ere dedignetur; an
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49 quia, c&ugrave;m inter c&aelig;teras $cientias Geometria demon$trationem,
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50 hoc e$t veritatem $inceram, &amp; quandam primi veri particu-
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51 lam profiteatur, hinc ne$cio quid diuinum habent $ibi propo$it&utilde;,
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52 vnde nonni$i Deo impellente, vbi nimirum officia, potiorque
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53 ratio id po$tulant, ab eius intuitu retrahatur. Hoc equidem
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54 puto; atque hinc diuina Geometria iure optimo a docti$$imis, &amp;
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55 clari$$imis viris pa$$im nuncupatur. Quamobrem nemo non
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56 eam <*>u$piciat, eiu$que cultores oppid&ograve; diligat; ob eamque caus&atilde;
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57 huic etiam qualicunque opu$culo benign&egrave; annuas $pero, adeo
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58 vt iam Te in terris Dominum, Altorem, Seruatorem, Patro-
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59 numque appellare non dubitem, quam vna cum Cel$i$$ima do-
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60 mo mihi, tot tibi nominibus deuincto, $uperi vt $eruent $o$pi-
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61 tentque, enix&egrave; oro, ac omnibus votis exopto.</I>
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62 <p><I>Sereni$sim&aelig; Cel$itudinis Tu&aelig;</I>
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63 <p><I>Humillimus, &amp; Ob$equenti$$imus Seruus</I>
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64 <p>Ioannes Ceua.
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65 <pb n=1>
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66 <C>GEOMETRIA</C>
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67 <C>MOTVS.</C>
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68 <C>DEF. I.</C>
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69 <p>CVrrat mobile ab A in D $ecund&ugrave;m rectam
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70 <MARG><I>Tab.</I> 1. <I>Fig.</I> 1.</MARG>
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71 AD, &amp; linea BHI $it natur&aelig; illius, vt dedu-
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72 ctis ad AD perpendicularibus AB, CH, DI
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73 ex punctis quibu$cunque A, C, D; veloci-
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74 tatum gradus, quos mobile $ortitur in ij$-
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75 dem punctis A, C, D men$urentur ab ip$is
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76 rectis AB, CH, CI. Figuram planam BADIHB apellabi-
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77 mus gene$im motus ab A in D.
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78 <C>DEF. II.</C>
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79 <p>II$dem manentibus, $it etiam alia linea EFG talis natu-
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80 <MARG><I>Tab.</I> 1. <I>Fig.</I> 2.</MARG>
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81 r&aelig;, vt protractis rectis BA in E, HC in F, &amp; ID in G ha-
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82 beat DG ad CF eandem reciproc&egrave; rationem, quam HC
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83 ad ID. Item $it CF ad HE vt reciproc&egrave; BA ad HC, vo-
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84 cabimus figuram planam ADGIEA imaginem tempo-
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85 ris motus ab A in D iuxta gene$im pr&aelig;dictam.
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86 <C>DEF. III.</C>
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87 <p>ADhuc po$ita illa gene$i, intelligatur linea PON eius
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88 <MARG><I>Tab.</I> 1. <I>Fig.</I> 3.</MARG>
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89 natur&aelig;, vt $i $it KL ad LM vt tempus lationis ab A
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90 in C ad tempus ab eodem C in D, habeat $emper KP ad
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91 LO eandem rationem, quam AB ad CH; &amp; LO ad NM
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92 eandem, quam HC ad ID: Figuram planam PKMNOP
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93 <pb n=2>
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94 vocabimus imaginem iuxta gene$im BADI motus ab
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95 A in D.
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96 <C><I>Corollarium.</I></C>
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97 <p><I>Patet, cum motus $unt &aelig;quabiles, gene$es, &amp; imagines figu-
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98 ras e&szlig;e parallelogrammas.</I>
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99 <C>DEF. IV.</C>
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100 <MARG><I>Tab.</I> 1. <I>Fig.</I> 4.</MARG>
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101 <p>SI $int du&aelig; gene$es, aut imagines ABCD, FEG, ita vt
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102 cum gene$es $int, habeat AB ad FE eandem rationem,
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103 quam velocitas in A ad velocitatem in F, &amp; cum imagines
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104 velocitatum, quarum tempora AD, FG, velocitas, quam
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105 habet mobile in$tanti A ad velocitatem alterius mobilis
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106 in$tanti F, $it vt AB ad FE, &amp; demum ip$is figuris vt imagi-
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107 nibus temporum con$ideratis habeat velocitas in A ad
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108 velocitatem in F rationem eandem, quam AB ad FE, vo-
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109 cabuntur tum gene$es ill&aelig;, cum imagines inter $e homo-
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110 gene&aelig;.
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111 <C>DEF. V.</C>
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112 <p>EAm planam Figuram, in qua duct&aelig; quoteunque
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113 &ecedil;quidi$tantes e&ograve; deinceps decre$cunt, qu&ograve; ad idem
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114 extremum propiores fiunt, acuminatam nuncupabimus.
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115 <C>DEF. VI. AX. I.</C>
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116 <p>INter maximam, &amp; minimam eiu$dem imaginis veloci-
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117 tatem cadit qu&aelig;dam media, qua tant&ugrave;m velocitate, $i
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118 conciperetur motus &aelig;quabilis, nihilomin&ugrave;s eodem tem-
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119 pore idem $patium curreretur, ac iuxta imaginem propo$i-
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120 tam: eam ergo mediam velocitatem dicimus propo$it&aelig;
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121 imaginis &aelig;quatricem.
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122 <pb n=3>
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123 <C>AX. II.</C>
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124 <p>SPatium iuxta imaginem velocitatum quamcunque
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125 exactum, vel iuxta &aelig;quatricem imaginis e$t maius eo
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126 $patio, quod curreretur eodem tempore minima eiu$dem
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127 imaginis velocitate; $ed minus eo, quod velocitate ma-
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128 xima.
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129 <C>AX. III.</C>
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130 <p>TEmpus, quo curritur $patium iuxta quamlibet tem-
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131 poris imaginem, maius e$t eo, quo idem $patium
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132 curreretur maxima velocitate, $ed contra minus eo altero,
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133 quo ip$um $patium minima velocitate exigeretur, earum
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134 videlicet, qu&aelig; $unt in gene$i, aut imagine velocitatum pro-
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135 po$iti motus, cuius nempe illa e$t imago temporis. Fit er-
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136 go, vt tempus &aelig;quale ei, quo illud ip$um $patium currere-
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137 tur iuxta propo$itam imaginem, $it inter vtrumque dicto-
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138 rum temporum maximi, &amp; minimi.
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139 <C>AX. IV.</C>
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140 <p>QV&aelig;cunque excogitetur figura plana, vel e$t paralle-
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141 logrammum, vel acuminata figura, aut ex his com-
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142 po$itum. Has tamen figuras inter binas volu-
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143 mus parallelas, ita vt vnum latus $it ip$as nectens normali-
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144 ter parallelas, quanquam etiam loco parallelarum po$$int
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145 e$$e puncta, nemp&egrave; vbi de$inunt in acuminatas pror$us
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146 figuras.
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147 <C>PROP. I. THEOR. I.</C>
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148 <p>TEmpora, quibus duo motus complentur $unt in ra-
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149 <MARG><I>Tab.</I> 1. <I>Fig.</I> 5.</MARG>
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150 tione imaginum homogenearum ip$orum tempor&utilde;.
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151 <pb n=4>
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152 <p>Motus $int prim&ograve; &aelig;quabiles, curratque mobile $patium
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153 AB tempore, cuius imago CAB, curratur item ab alio mo-
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154 bili $patium DE tempore, cuius imago DEF, &amp; $int ip$&aelig;
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155 <MARG><I>Def.</I> 4. <I>huius.</I></MARG>
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156 temporum imagines inter$e homogene&aelig;, $cilicet FD ad
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157 AC eandem habeat rationem, quam velocitas in A ad
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158 velocitatem in D. Dico, tempus per AB ad id per DE e$-
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159 <MARG><I>Cor. Def.</I> 3.
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160 <I>huius.</I></MARG>
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161 $e vt figura ABC, ad DEF. Cum motus &aelig;quabiles $int
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162 erunt figur&aelig; dictarum imaginum rectangula, propterea il-
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163 lorum ratio componetur ex rationibus altitudinum AB ad
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164 <MARG><I>Gal. pr. S de
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165 motu &aelig;quab.
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166 Def.</I> 4. <I>huius.</I></MARG>
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167 DE, &amp; ba$ium AC ad DF, ex ij$dem ver&ograve; rationibus $pa-
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168 tiorum $cilicet, &amp; reciproca velocitatum ($unt enim ima-
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169 gines inter $e homogene&aelig;) nectitur etiam ratio temporum,
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170 quibus percurr&utilde;tur ip$a $patia AB, DE iuxta gene$es ima-
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171 ginum ACB, DEF, ergo e$t eadem ratio inter illa tempo-
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172 ra, ac inter imagines $uas.
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173 <MARG><I>Tab.</I> 1. <I>fig.</I> 6.
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174 <I>Def.</I> 5. <I>huius.</I></MARG>
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175 <p>2. Sit motus vnus &aelig;quabilis, alter ver&ograve; quicunque; $it
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176 tamen imago huius temporis figura acuminata vt ALGE,
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177 &amp; alterius temporis pr&aelig;dicti motus &aelig;quabilis, $it HFM,
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178 <MARG><I>Cor. Def.</I> 3.
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179 <I>huius.</I></MARG>
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180 qu&aelig; rectangulum erit: Dico, imaginibus homogeneis exi-
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181 $tentibus, fore inter has eandem rationem, ac homolog&egrave;
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182 inter tempora decur$uum ab A in E, &amp; ab F in M iuxt&atail;
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183 gene$es imaginum temporum propo$itarum. Si enim non
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184 e$t ita, $it qu&aelig;dam alia magnitudo Y, maior, vel minor
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185 imagine acuminata ALGE, qu&aelig; ad imaginem FHM ha-
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186 beat eandem rationem, quam tempus per AE iuxta imagi-
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187 nem ALGE ad tempus per FM iuxta imaginem alteram
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188 FHM; $it ver&ograve; magnitudinis Y differentia ab imagine ma-
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189 gnitudo Z. Secetur AE bifariam in C, pariterque $eg-
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190 menta AC, CE bifariam in B, D, &amp; $ic vlteri&ugrave;s progredia-
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191 tur, donec, $i compleatur rectangulum po$tremum, &amp; ma-
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192 ximum DG, hoc minus exi$tat quam Z. Tum ductis reli-
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193 quis &aelig;quidi$tantibus CI, BK, &amp; &agrave; punctis N, <*>, K, I alijs
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194 etiam &aelig;quidi$tantibus rect&aelig; AE, efficiatur ip$i ALGE cir-
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195 cum$cripta figura, con$tans ex rectangulis &aelig;quealtis AK
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196 <pb n=5>
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197 BI, CN, DG, &amp; in$cripta compo$ita ex rectangulis inter $e
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198 pariter &aelig;quealtis BL, CR, DI, EN. Cum circum$cript&atail;
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199 figura differat ab in$cripta exce$$u, quo rectangulum DG
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200 $uperat BL; (nam reliqua circum$cripta AK, BI, CN, re-
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201 liquis in$criptis &aelig;qualia $unt) $e quitur, exce$$um illum e$$e
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202 minorem magnitudine Z. Si ergo magnitudo Y ponatur
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203 maior magnitudine ALGE pro exce$$u Z, maior etiam erit
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204 circum$cripta AK, BI, CN, DG. Qu&ograve;d $i contr&agrave; Y intelli-
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205 gatur minor ip$a ALGE ex defectu Z, erit quoque eadem
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206 Y minor, qu&agrave;m in$cripta figura BL, CK, DI, EN. Itaque
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207 nunc, $i fieri pote$t, $it Y maior magnitudine ALGE per ip-
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208 $um exce$$um Z, &amp; intelligantur tot motus, quot $unt re-
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209 ctangula in circum$cripta figura, $cilicet $int ip$i motus ab
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210 A in B, &agrave; B in C, &agrave; C in D, &amp; &agrave; D in E $ecundum deinceps,
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211 temporum imagines AK, BI, CN, DG rectangula, qu&aelig;
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212 $int inter$e, &amp; propo$itis imaginibus homogene&aelig;, qui
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213 motus erunt proptere&agrave; &aelig;quabiles. His po$itis, tempus
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214 <MARG><I>Cor. Def.</I> 3.
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215 <I>huius.</I></MARG>
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216 per FM iuxta imaginem MH ad tempus per AB iuxta ima-
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217 ginem rectangulum AK eandem habet rationem, quam re-
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218 ctangulum MH ad rectangulum AK, $imiliter idem tem-
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219 pus per FM $ecund&ugrave;m ip$am imaginem rectangulum MH
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220 <MARG><I>Ex prami&szlig;&atail;
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221 parte.</I></MARG>
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222 ad $ingula reliqua tempora per BC, CD, DE imaginibus
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223 deinceps rectangulis BI, CN, DG habet eandem rationem,
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224 quam rectangulum MH ad $ingula eodem ordine rectan-
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225 gula BI, CN, DG. Quo circa totidem rectangula ex MH,
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226 <MARG><I>Euang. Tor-
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227 ric. lem.</I> 18. <I>in
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228 libro de dim.
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229 parabol&aelig;.</I></MARG>
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230 quot $unt illa, ex quibus con$tat circum$cripta figura, ha-
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231 bebunt ad ea ip$a circum$cripta rectangula, $eu ad eandem
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232 circum$criptam figuram AK, BI, CN, DG eandem ratio-
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233 nem, quam totidem tempora eiu$dem imaginis MH ad $i-
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234 mul tempora, quorum imagines $unt illa ip$a circum$cripta
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235 rectangula AK, BI, CN, DG. Quare etiam vnicum re-
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236 ctangulum MH ad circum$criptam figuram AK, BI, CN,
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237 DG erit in eadem ratione, in quo vnicum tempus per FM
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238 iuxta imaginem MH ad omnia $imul illa tempora iuxt&atail;
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239 <pb n=6>
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240 imagines, qu&aelig; $unt dicta circum$cripta rectangula. Et
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241 quoniam figura imaginis e$t acuminata, habetque vi def.
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242 2. huius, applicatas, qu&aelig; $unt in ratione reciproca veloci-
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243 tatum, quibus nempe mobile afficitur in punctis $patij, &agrave;
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244 quibus deducuntur ip$&aelig; applicat&aelig;; hinc fit, vt earum ve-
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245 locitatum, quas mobile habet in decur$u rect&aelig; AB, ea, qu&ecedil;
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246 in A maxima $it, &amp; qu&aelig; in B minima. Eodem modo iuxta
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247 reliquas imagines BKIC, CIND, DNGE, qu&aelig; itidem acu-
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248 minat&aelig; $unt, velocitates in fine decur$uum C, D, E ($unt
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249 enim omnes vers&ugrave;s A acuminat&aelig;) minim&aelig; erunt, &amp; ma-
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250 xim&aelig; initio dictorum $patiorum. Ideo tempora, qu&ecedil; im-
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251 <MARG><I>Ax.</I> 3. <I>huius.</I></MARG>
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252 penduntur iuxta illas imagines, $eu ip$am imagin&etilde; ALGE,
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253 cuius ill&aelig; $unt omnes partes, minora erunt temporibus,
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254 qu&aelig; decurrerent, $i illi decur$us forent &aelig;quabiles ex mini-
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255 mis illis velocitatibus exacti, vel quod in idem recidit, $i
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256 illi decur$us e$$ent iuxta imagines rectangulorum circum-
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257 $criptorum AK, BI, CN, DG; itaque rectangulum MH ad
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258 figuram circum$criptam AK, BI, CN, DG habebit mino-
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259 rem rationem, qu&agrave;m tempus per FM imagine MH ad tem-
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260 pus per AE imagine ALGE, $eu qu&agrave;m rectangulum MH
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261 habet ex hypothe$i ad magnitudinem Y; igitur circum$cri-
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262 pta figura, qu&aelig; pri&ugrave;s minor o$ten$a fuit magnitudine Y;
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263 nunc maior concluditur; quod cum $it ab$urdum, $equi-
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264 tur fals&ograve; nos po$ui$<*>e magnitudinem Y maiorem; qu&agrave;&mtail;
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265 ALGE. At $i Y minor ponatur, qu&atilde; magnitudo ALGE de-
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266 fectu Z; in$cripta, vt $upra, figura con$tante ex rectangulis
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267 &aelig;qu&egrave; altis BL, CK, DI, EN, vt $cilicet differentia ab ima-
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268 gine $it minor magnitudine Z, liquebit, magnitudinem Y
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269 minorem e$$e in$cripta figura BL, CK, DI, EN; deind&etail;
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270 procedendo vt $upra, inueniemus rectangulum MH ad in-
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271 $criptam figuram BL, CK, DI, EN in eadem ratione, i&ntail;
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272 quo tempus per FM imagine MH ad omnia $imul decur-
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273 $uum tempora per AB, BC, CD, DE iuxta imagines re-
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274 ctangula in$cripta BL, CH, DI, EN; H&aelig;c ver&ograve; tempor&atail;
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275 <pb n=7>
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276 minora $unt temporibus iuxta imagines ALKB, BKIC,
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277 CIND, INGE (nam velocitates initio decur$uum per
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278 dictas rectas diximus e$$e maximas, &amp; quibus con$ider&atilde;-
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279 tur illi motus &aelig;quabiles $ecund&ugrave;m imagines ip$a illa re-
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280 ctangula in$cripta) ergo rectangulum MH ad in$cripta&mtail;
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281 figuram BL, CK, DI, EN habebit maiorem rationem, qu&atilde;
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282 tempus per FM iuxta imaginem MH ad tempora $imul
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283 imaginibus ALKB, BKIC, CIND, DNGE, $iue ad tempus
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284 iuxta imaginem ALGE ex illis compo$itam. Ideoque re-
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285 ctangulum MH ad ip$am in$criptam figuram habebit ma-
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286 iorem rationem, qu&agrave;m ad magnitudinem Y, idcirco Y, qu&aelig;
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287 minor o$ten$a fuit in$cript&agrave; figura BL, CK, DI, EN, nunc
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288 hac alia via maiorem inuenimus; ergo cum rur$us hoc $it
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289 ab$urdum, nece$$e e$t magnitudinem Y neque minore&mtail;
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290 e$$e magnitudine ALGE, propterea &aelig;quales inter $e er&utilde;t,
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291 atque adeo tempus per FM imagine MN ad tempus per
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292 AE imagine ALGE habebit eandem rationem, quam ima-
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293 go MH ad imaginem ALGE. Quod &amp;c.
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294 <p>3. Imagines propo$it&aelig; $int du&aelig; acuminat&aelig;. Dico ni-
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295 <MARG><I>Tab.</I> 1. <I>Fig.</I> 7.</MARG>
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296 hilominus, tempora iuxta illas imagines per AE, HI e$$e vt
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297 ip$&aelig; imagines ALGE ad HIK, qu&aelig; $int inter $e homoge-
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298 ne&aelig; vt $emper $upponetur. Nam $i intelligatur alius mo-
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299 tus per MF iuxta imaginem rectangulum MFN, qui &aelig;qua-
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300 <MARG><I>Cor. Def.</I> 3
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301 <I>huius.</I></MARG>
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302 bilis erit, manife$tum e$t ex $ecundo ca$u, tempus per AE
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303 iuxta imaginem ALGE ad tempus per FM iuxta imagin&etilde;
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304 rectangulum MH, habere eandem rationem, quam imago
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305 ALGE ad imaginem rectangulum MH; &amp; $imiliter tem-
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306 pus per FM imagine rectangulum MN ad tempus per HI
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307 iuxta imaginem HKI habet eandem rationem, quam ima-
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308 go NM ad imaginem HKI, ergo ex &aelig;quali tempus per AE
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309 ad tempus per HI $ecund&ugrave;m imagines propo$itas erit vt
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310 imago ip$a ALGE ad imaginem HKI. Quod &amp;c.
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311 <p>4. Demum imagines $int qu&aelig;cunque, mod&ograve; $int ho-
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312 <MARG><I>Tab.</I> 1 <I>fig. <*></I></MARG>
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313 mogene&aelig;, ADFB, GHKL: Dico rur$us inter $e e$$e vt tem-
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314 <pb n=8>
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315 pora per AB, AK iuxta ip$a imagines. Vel enim h&aelig; ima-
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316 gines $unt $implices, hoc e$t tant&ugrave;m parallelogramm&ecedil;, aut
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317 tant&ugrave;m acuminat&aelig;, &amp; tunc $upra o$tendimus propo$itum,
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318 quemadmodum etiam $i vna acuminata, altera parallelo-
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319 gramma; vel non $unt huiu$modi &amp; componentur ex illis.
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320 <MARG><I>Ax.</I> 4. <I>huius.</I></MARG>
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321 Sint ergo in imagine ADFB partes ab &aelig;quidi$tantibus di-
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322 $tinct&aelig; ADEN, OFB acuminat&aelig; &amp; NEFO paralellogram.
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323 <MARG><I>Def.</I> 4. <I>huius.</I></MARG>
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324 mum, erunt h&aelig; procul dubio inter $e, totique imagini ho-
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325 mogene&aelig;; $int pariter in alia imagine partes GHCM,
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326 MCKL, per &aelig;quidi$tantem MC di$tinct&aelig; inter $e acumi-
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327 <MARG><I>Def:</I> 4. <I>huius.</I></MARG>
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328 nat&aelig;, qu&aelig; itidem inter $e, &amp; imagini, cuius $unt partes, ho-
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329 mogene&aelig; erunt. His acceptis, quoniam tempus per AN
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330 <MARG><I>Ex tertia
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331 parte huius.</I></MARG>
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332 iuxta imaginem ADEN acuminatam ad tempus per HC
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333 iuxta aliam imaginem item acuminatam HGMC, habet
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334 eandem rationem, ac imago ADEN ad imagin&etilde; GHCM.
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335 $imiliter tempus per HC iuxta imaginem GHCM ad tem-
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336 pus per CK iuxta imaginem acuminatam MCKL e$t vt
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337 illa ad hanc imaginem; componendo, inde per conuer$io-
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338 nem rationis, &amp; conuertendo, tempus per HC $ecund&ugrave;m
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339 imaginem GHCM ad tempora $imul per HC, CK, quor&utilde;
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340 imagines GHCM, MCKL, hoc e$t ad tempus per HK iux-
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341 ta imaginem GHKL habebit e&atilde;dem rationem, quam ima-
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342 go GHCM ad imaginem GHCL; &amp; ideo ex &aelig;quali tem-
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343 pus per AN, cuius imago ADEN, ad tempus per HK, iux-
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344 ta imaginem GHKL, erit in eadem ratione, in qua e$t ima-
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345 go ADEN ad imaginem GHKL. Pr&aelig;terea tempus per
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346 AN iuxta imaginem ADEN ad idem ip$um tempus habet
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347 eandem rationem, quam imago ADEN ad eandem ip$am;
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348 tempus per NO iuxta imaginem rectangulum NEPO ad
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349 <MARG><I>Ex</I> 2. <I>part&etail;
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350 huius.</I></MARG>
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351 tempus pr&aelig;dictum per AN e$t in eadem ratione imagin&utilde;
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352 NEPO ad ADEN, &amp; $imiliter tempus per OB iuxta ima-
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353 ginem OPFB habet ad tempus per AN eandem rationem,
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354 ac imago OPFB ad imaginem $&aelig;p&egrave; dictam ADEN; itaq; ex
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355 lem. 18. Toric. in lib. de dim: parabol&aelig;, erunt tria t&etilde;pora per
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356 <pb n=9>
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357 AN, NO, OB iuxta imagines deinceps ADEN, NEPO,
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358 OPFB, hoc e$t erit tempus per AB iuxta imaginem ADFB
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359 ad $imul tria tempora per AN iuxta eandem imaginem
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360 ADEN, vt imago ADFB ad triplum imaginis ADEN, &amp;
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361 cum tria &aelig;qualia tempora per AN ad vnicum ex illis $it
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362 vt triplum imaginis ADEN ad vnicam imaginem; $equi-
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363 tur ex &aelig;quali tempus per AB ad tempus per AN iuxt&atail;
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364 imaginem ADEN habere eandem rationem, quam imago
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365 ADFB ad imaginem ADEN: &amp; o$ten$um fuit tempus per
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366 AN iuxta imaginem ADEN ad tempus per HK iuxta
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367 imaginem GHKL habere eandem rationem, quam imago
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368 ADEN ad imaginem GHKL, ergo rur$us, &amp; tandem ex
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369 &aelig;quali, tempus per AB iuxta imaginem ADFB ad t&etilde;pus
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370 per HK iuxta imaginem GHKL habebit eandem ration&etilde;,
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371 quam imago ADFB ad imaginem GHKL. Quod &amp;c.
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372 <C><I>Corollarium.</I></C>
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373 <p><I>Hinc colligitur, $i prima magnitudo ad $ecundam fuerit vt
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374 tertia ad quartam, item alia prima ad aliam $ecundam vt
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375 alia tertia ad aliam quartam, &amp; $ic vlteri&ugrave;s quoad vi$u&mtail;
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376 fuerit, $int pr&aelig;terea omnes prim&aelig;, item omnes terti&aelig; inter$e
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377 &aelig;quales, con$tat, inquam, primarum vnam ad omnes $ecun-
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378 das habere eandem rationem, ac vna tertiarum ad omnes
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379 quartas.</I>
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380 <C>PROP. II. THEOR. II.</C>
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381 <p>SSpatia, qu&aelig; curruntur iuxta qua$cunque homogeneas
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382 velocitat&utilde; imagines, $unt inter$e, vt e&aelig;dem ill&aelig; ima-
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383 gines. Sint prim&ugrave;m motus &aelig;quabiles, curraturque $pa-
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384 <MARG><I>Tab.</I> 1. <I>Fig.</I> 9.
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385 <I>Cor. Dif.</I> 3.
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386 <I>huius.</I></MARG>
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387 tium AB iuxta imaginem velocitatum, qu&aelig; rectangulum
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388 erit ILMK, $patium ver&ograve; DE tran$igatur iuxta imagine&mtail;
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389 pr&aelig;dict&aelig; homogeneam rectangulum FHNG (nam erunt
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390 <pb n=10>
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391 homogene&aelig; ip$&aelig; imagines, $i vt ex Def. 4. huius IL ad HF
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392 erit vt velocitas in$tanti I ad velocitatem mobilis in$tanti
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393 F) Dico $patium AB ad DE e$$e vt imago rectangulu&mtail;
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394 ILMK ad imaginem rectangulum FHNG. Componuntur
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395 ip$a illa rectangula ex ratione altitudinum IK ad FG, &amp; ex
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396 ea ba$ium IL ad FH; ver&ugrave;m ex ij$dem, ea nempe tempor&utilde;
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397 <MARG><I>Gil. de motu
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398 &aelig;quabili.</I></MARG>
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399 IK ad FG, atque ea velocitatum IL ad FH componitur
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400 etiam ratio $patiorum AB ad DE, ergo ip$a $patia erunt vt
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401 propo$it&ecedil; imagines.
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402 <MARG><I>Tab.</I> 1. <I>fig</I> 10.</MARG>
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403 <p>2. Sint nunc motus iuxta imagines, quarum altera acu-
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404 minata, altera rectangulum $it. Dico rur$us $patium AB,
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405 quod curritur iuxta imaginem ABCD ad $patium DE,
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406 quod curritur iuxta alteram imaginem, e$$e vt imago
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407 ABCD ad imaginem PHNG. Ni$i ita $it, erit alia magni-
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408 tudo Y maior, vel minor imagine ABCD, qu&aelig; quidem ad
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409 alteram imaginem HPGN habebit eandem rationem, qu&atilde;
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410 $patium AB ad DE. Sit prim&ugrave;m maior exce$$u Z. Cir-
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411 cum$cribatur; vt egimus in $ecunda parte prim&aelig; huius, fi-
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412 gura imagini ABCD con$tans ex rectangulis &aelig;qu&egrave; altis,
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413 excedatque imaginem ABCD exce$$u minori, quam Z; $it
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414 ergo circum$cripta illa AE, HF, IG, KG, quam prim&ograve; fa-
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415 cil&egrave; o$tendemus minorem magnitudine Y; nam h&aelig;c exce$-
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416 $u magis di$tat ab imagine, qu&agrave;m circum$cripta illa. Pr&aelig;-
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417 terea $i intelligantur tot motus &aelig;quabiles, quot $unt rect&atilde;-
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418 gula circum$cripta, ij nempe, qui fierent temporibus AH,
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419 HI, IK, KD iuxta deinceps imagines ip$a rectangula AE,
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420 HF, IG, KC inter$e, &amp; propo$itis imaginibus homogeneas,
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421 velocitates, quibus ijdem motus con$iderarentur, forent
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422 HE, IF, KG, DC, nimirum maxim&aelig; imaginum ABEH,
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423 HEFI, IFGK, KGCD; Cumque ita $it, longiora $patia cur-
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424 <MARG><I>Ax.</I> 2. <I>huius.</I></MARG>
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425 rerentur iuxta imagines rectangula circum$cripta, quam
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426 ij$dem temporibus, imaginibu$que po$tremis, hoc e$t qu&atilde;
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427 tempore AD iuxta imaginem ABCD; obidque $patium
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428 AB ad DE, $eu magnitudo Y ad imaginem HPGN habe-
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429 <pb n=11>
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430 bit minorem rationem, qu&agrave;m omnes ill&aelig; $imul imagines,
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431 <MARG><I>Cor. pr.</I> 1. <I>hu-
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432 ius.</I></MARG>
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433 $eu quam circum$cripta figura AE, HF, IG, KC ad ean-
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434 dem imaginem HPGN; quare Y, qu&aelig; pri&ugrave;s o$ten$a fuit
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435 maior, nunc reperitur minor eadem circum$cripta, quod
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436 cum fieri nequeat, impo$$ibile etiam e$t magnitudinem Y
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437 maiorem e$$e magnitudine imaginis ABCD. Sit ergo mi-
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438 nor, $i etiam fieri pote$t, &amp; defectus ip$ius Y $upra ABCD
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439 $it Z. In$cribatur imagini figura ex rectangulis &aelig;quealtis, vt
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440 nempe deficiat ab imagine defectu minori Z; $ic enim ip$a
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441 in$cripta, qu&aelig; $it AB, IE, KF, DG erit magnitudine pro-
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442 pinquior imagini ABCD, qu&agrave;m Y, ideoque Y minor erit
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443 dicta in$cripta figura. Deinde, quoniam, $i ponantur mo-
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444 tus &aelig;quabiles, quorum imagines rect angula in$cripta HB,
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445 IE, KF, DG, qu&aelig;que inter $e, &amp; propo$itis imaginibus $int
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446 homogene&aelig;; velocitates, quibus efficerentur dicti motus,
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447 e$$ent AB, IE, KF, DG, minim&aelig; $cilicet imaginum ABEH
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448 HEFI, IFGK. KGCD, &amp; ideo $patia, qu&aelig; percurrerentur
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449 temporibus HA, HI, IK, KD imaginibus illis, maiora e$-
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450 <MARG><I>Ex.</I> 2 <I>huius.</I></MARG>
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451 $ent, qu&agrave;m qu&aelig; ij$dem temporibus tran$igerentur iuxt&atail;
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452 imagines pr&aelig;dictas rectangula circum$cripta, hinc fit vt
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453 $patium AB ad DE, $eu magnitudo Y ad imagine HPGN
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454 habeat maiorem rationem, qu&agrave;m in$cripta figura ad ean-
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455 dem imaginem HPGN; quare Y, qu&aelig; minor erat in$cripta
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456 figura, mod&ograve; re$ultat maior, non ergo Y minor e$$e pote$t
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457 imagine ABCD, $ed neque maior vt o$tendimus, ergo $pa-
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458 tium AB ad DE erit, vt imago ABCD ad imaginem
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459 PHNG. Quod &amp;c.
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460 <p>3. &amp; 4. Si ver&ograve; imagines acuminat&aelig; $int, aut demum
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461 qu&aelig; cumque, eodem prors&ugrave;s modo, quo prima propo$itio-
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462 ne, o$tendemus hoc etiam propo$itum, ergo patet omne
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463 intentum.
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464 <pb n=12>
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465 <C><I>Scholium.</I></C>
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466 <p><I>Cum prors&ugrave;s geometric&egrave; o$tenderimus $uperiores duas pro-
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467 po$itiones, vtili$$imum e$t ob$eruare, quomodo liceat vti tem-
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468 poris in$tantibus, non vt punctis prors&ugrave;s geometricis, $ed vt
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469 quantitatibus dicam minoribus quibu$cunque datis. Hinc
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470 oritur indiui$ibilium method<*>s, qu&aelig; intelligentiam affert
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471 faciliorem, ac $i rigori geometrico penitus in$i$teremus, quam-
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472 quam e&aelig; tamen difficiliores Geometras mihi magis decer&etail;
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473 videantur.</I>
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474 <C>PROP. III. THEOR. III.</C>
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475 <MARG><I>Tab.</I> 2, <I>Fig.</I> 1.</MARG>
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476 <p>SPatia, qu&aelig; curruntur iuxta quaslibet homogeneas ve-
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477 locitatum imagines, nectuntur ex rationibus tempo-
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478 rum, ac &aelig;quatricum.
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479 <p>Velocitates &aelig;quatrices duorum motuum, quorum ima-
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480 gines velocitatum $int ABCD, EFHI ponantur AG, EL.
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481 Dico $patia, $eu ip$as imagines componi ex ratione tem-
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482 porum AD ad EI; &amp; ex ea &aelig;quatricum AE ad EL. Nam
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483 $i motus, qui e$t iuxta imaginem ABCD per$eueret velo-
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484 citate AG, e$$et quidem &aelig;quabilis, idemque $patium illa
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485 <MARG><I>Def.</I> 6. <I>Ax.</I> 1.</MARG>
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486 velocitate, &amp; tempore AD percurreretur, ac $ecund&ugrave;&mtail;
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487 imaginem ABCD; Itaque exi$tente rectangulo DE, quod
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488 <MARG><I>Cor.</I> 3. <I>Def.</I> 3.
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489 <I>huius.</I></MARG>
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490 e$set imago velocitatum illius motus &aelig;quabilis, foret idem
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491 <MARG><I>Pr.</I> 2. <I>huius.</I></MARG>
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492 &aelig;quale imagini ABCD (nam imagines ABCD, &amp; DG
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493 homogene&aelig; $unt) eodem modo imago rectangulum VL
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494 &aelig;quale e$set imagini EFHI. Cum ergo du&aelig; imagines re-
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495 ctangula DE, IL componantur ex rationibus temporum
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496 AD ad EI, &amp; ex ea &aelig;quatricum AG ad EL; ex ij$de&mtail;
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497 prors&ugrave;s rationibus etiam imagines propo$it&aelig; pr&aelig;dictis re-
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498 ctangulis &aelig;quales nectentur. Et ideo $patia, qu&aelig; propo-
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499 $itis imaginibus tran$iguntur, qu&aelig;que ip$is proportionalia
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500 <pb n=13>
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501 $unt, componentur ex rationibus temporum, &amp; ex rationi-
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502 bus &aelig;quatricum.
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503 <C><I>Corollarium I.</I></C>
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504 <p><I>Hinc patet $i line&aelig;, qu&aelig; in imagine velocitatum tempus
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505 ex hibet, aplicetur rectangulum &aelig;quale propo$it&aelig; imagini ve-
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506 locitatum, fore vt latitudo eiu$dem rect anguli, $it velocitas
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507 &aelig;quatrix propo$it&aelig; imaginis.</I>
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508 <C><I>Corollarium II.</I></C>
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509 <p><I>Item const at, vbi tempora, vel &aelig;quatrices velocitates fue-
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510 rint &aelig;quales, rationem $patiorum e$$e eandem, qu&aelig; &aelig;quatri-
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511 cum, vel qu&aelig; temporum.</I>
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512 <C><I>LEMMA.</I></C>
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513 <p><I>Si qu&aelig;libet ratio compo$ita $it ex quotcumque rationibus,
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514 harum qu&aelig;libet nectetur ex propo$ita, &amp; ex reliquis contra-
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515 ri&ograve; $umptis rationibus. Sit A ad B compo$ita ex rationibus E
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516 &aelig;d F; G ad H; &amp; I ad K. Dico quamlibet ist arum puta G ad
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517 K con$tare ex rationibus A ad B, &amp; ex reliquis reciproc&egrave; $um-
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518 ptis F ad E, &amp; I ad K. Vt E ad F, ita $it A ad C, &amp; vt D ad B
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519 $ic I ad K; erit C ad D, vt G ad H; ideoq; C ad D, hoc e$t G ad</I>
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520 <TABLE>
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521 <TR><TD><I>A</I></TD><TD><I>E</I></TD><TD></TD><TD></TD></TR>
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522 <TR><TD><I>C</I></TD><TD><I>F</I></TD><TD><I>I.</I></TD><TD><I>K</I></TD></TR>
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523 <TR><TD><I>D</I></TD><TD><I>G</I></TD><TD></TD><TD></TD></TR>
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524 <TR><TD><I>B</I></TD><TD><I>H</I></TD><TD></TD><TD></TD></TR>
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525 </TABLE>
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526 <I>H nectetur ex C ad A, $eu F ad G, &amp; ex rationibus A ad B,
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527 B ad D, $iue K ad I. Quod &amp;c.</I>
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528 <pb n=14>
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529 <C>PROP. IV. THEOR. IV.</C>
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530 <p>TEmpora, quibus ab$oluuntur duo motus componun-
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531 tur ex ratione $patiorum, &amp; ex reciproca &aelig;quatri-
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532 cum. Cum enim $patia compon&atilde;tur ex ratione temporum,
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533 <MARG><I>Pr.</I> 3. <I>huius.</I></MARG>
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534 &amp; ex ea velocitatum &aelig;quatricum, $equitur per pr&aelig;dictum
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535 Lenima, qu&ograve;d tempora nectantur ex rationibus $patiorum,
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536 &amp; reciproca &aelig;quatricum.
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537 <C><I>Corollarium.</I></C>
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538 <p><I>Manife$tum e$t $patia, vel &aelig;quatrices velocitates, $i $int
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539 &aelig;quales, e$$e tempora in reliqua ratione reciproca &aelig;quatri-
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540 cum, vel $patiorum non reciproca.</I>
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541 <C>PROP. V. THEOR. V.</C>
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542 <p>&AElig;Quatrices velocitates componuntur ex rationibus
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543 $patiorum, &amp; reciproca temporum.
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544 <p>Cum $patia componantur ex rationibus temporum, &amp;
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545 velocitatum &aelig;quatricum, manife$tum e$t ex eodem Lem-
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546 mate, velocitates ip$as necti ex rationibus $patiorum, &amp;
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547 reciproca temporum.
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548 <C><I>Corollarium.</I></C>
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549 <p><I>Deducitur, &aelig;quatrices velocitates e$$e vt tempora reciproc&egrave;
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550 $umpta, vel vt $patia, $i altera ratio fuerit &aelig;qualitatis.</I>
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551 <C>D. E F. VII.</C>
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552 <MARG><I>Tab.</I> 2. <I>Fig.</I> 2.</MARG>
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553 <p>SI in gene$ibus homogeneis AEC, GFK exi$tente AB
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554 ad BC $icut GI ad IK, habeat AE ad BD eandem ra-
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555 <pb n=15>
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556 tionem, ac GF ad IH, motus, qui fiunt iuxta illas gene$es,
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557 vocentur inter $e $imiles, &amp; ip$&aelig; gene$es dicentur $imilium
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558 motuum; quod ver&ograve; attinet ad rectas AE, BD, GF, IH apel-
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559 labimus applicatas ad homologa puncta A, B, G, I propor-
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560 tionales.
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561 <C>PROP. VI. THEOR. VI.</C>
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562 <p>SI in imaginibus temporum homogeneis, applicat&aelig; v-
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563 nius fuerint ad homologa puncta, proportionales ap-
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564 plicatis alterius imaginis, motus, quorum $unt ip$&aelig; imagi-
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565 nes, $imiles erunt.
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566 <p>Imagines temporum $int &amp;MLABC, &amp;ONGIK, qu&aelig;
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567 <MARG><I>Tab.</I> 2. <I>fig.</I> 3.</MARG>
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568 $int homogene&aelig;, &amp; cum GI ad IK $it vt AB ad BC, habeat
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569 quoque AL ad BM eandem rationem, ac GN ad IO. Di-
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570 co, motus, quorum $unt ill&aelig; imagines temporum inter $e $i-
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571 miles e$$e.
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572 <p>Sint apud ip$as imagines eorundem motuum gene$es,
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573 $cilicet EAC, FGK inter$e homogene&aelig;. Exi$tente AL ad
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574 BM, vt GN ad IO, erit conuertendo BM ad AL vt IO ad
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575 GN; $ed vt BM ad AL ita ob gene$im EA ad DB, &amp; vt IO
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576 <MARG><I>Def.</I> 2. <I>huius.</I></MARG>
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577 ad GN, $ic FG ad HI. ergo EA ad DB e$t vt FG ad HI, erat
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578 autem vt AB ad BC ita etiam GI ad IK, ergo motus $unt
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579 <MARG><I>Def.</I> 7. <I>huius.</I></MARG>
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580 $imiles, &amp; ip$&aelig; imagines $imilium motuum.
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581 <C>PROP. VII. THEOR. VII.</C>
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582 <p>SI in imaginibus velocitatum vnius, applicate fuerint ex
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583 <MARG><I>Tab.</I> 2. <I>fig.</I> 4.</MARG>
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584 punctis homolog&egrave; $umptis proportionales applicatis
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585 alterius imaginis, motus iuxta ip$as imagines erunt $imi-
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586 les, ideoque ip$&aelig; imagines $imilium motuum.
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587 <p>Velocitatum imagines $int ABCD, NPRT, $itque AB
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588 ad EF in eadem ratione, in qua NP ad TR; Dico exi$tenti-
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589 bus etiam BF ad FC, vt PQ ad QR e$$e propo$itas imagi-
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590 nes $imilium motuum. Intelligantur eorundem motuum
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591 <pb n=16>
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592 gene$es GHKL, YZ 43. &amp; $it pariter HI ad IK, vt $egmen-
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593 tum ABFE ad EFCD. Sit $imiliter Z <*> ad <*> 4 vt $eg-
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594 mentum NPQV ad VQRT, ducti$que applicatis IM, QV,
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595 manife$tum e$t, vt velocitas AB &aelig;qualis e$t velocitati GH,
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596 $ic EF &aelig;qualem fore ip$i IM; nam quia $patium tran$act&utilde;
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597 iuxta imaginem ABFE ad $patium tran$actum imagine
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598 <MARG><I>Pr.</I> 2. <I>huius.</I></MARG>
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599 EFCD e$t vt illa ad hanc imaginem, nempe vt HI ad IK,
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600 erit mobile in$tanti F in puncto I, &amp; ideo inibi erit veloci-
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601 tas eadem, quam habet mobile in$tanti F, $cilicet &aelig;quales
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602 erunt EF, IM. Eodem modo erunt &aelig;quales QV, <*> 2, &amp;
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603 $unt etiam &aelig;quales NP, YZ, ergo $icut $e habet AB ad EF,
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604 ita erit GH ad MI, &amp; vt e$t NP ad VQ. ita erit YZ ad 2 <*>
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605 Pr&aelig;terea concipiatur figura OPRSXO $imilis ip$i ABCD,
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606 $cilicet $it CB ad PR vt AB ad OP, vel (cum $int BF ad
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607 FC ita PQ ad QR, vt EF ad homologam XQ, erit $eg-
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608 mentum ABFE ad $ibi $imile $egmentum OPQX in dupli-
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609 cata ratione laterum homologorum EF ad XQ, &amp; item in
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610 ead&etilde; duplicata ratione erunt inter$e $imilia $egm&etilde;ta EFCD
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611 ad XQRS, $ed cum etiam OPQX $egmentum ad NPQV,
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612 &amp; XQRS ad $egmentum VQRT $int in eadem ratione
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613 eiu$dem QX ad QV, erit ex &aelig;quali $egmentum ABFE ad
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614 $egmentum NPQV, vt $egmentum EFCD ad VQRT, &amp;
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615 permutando, $egmentum ABFE ad $egmentum EFCD ha-
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616 bebit eandem rationem, ac $egmentum NPQV ad VQRT
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617 $cilicet erit HI ad IK vt Z <*> ad <*> 4, ob idque con$tat ge-
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618 ne$ium applicatas vnius proportionales e$$e applicatis al-
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619 terius, quare $imiles motus erunt, qui fiunt iuxta imagines
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620 velocitatum propo$itas.
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621 <C>PROP. VIII. THEOR. VIII.</C>
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622 <p>SPatia, qu&aelig; curruntur $imilibus motibus $unt in ratione
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623 compo$ita temporum, &amp; homologarum velocitatum,
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624 interquas $unt extrem&aelig;, aut prim&aelig;.
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625 <pb n=17>
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626 <p>Imagines velocitatum $imilium motuum $int BCDE,
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627 <MARG><I>Tab.</I> 2. <I>Fig.</I> 5</MARG>
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628 GMKI, &amp; iuxta eas percurrantur $patia A, F. Dico i$ta com-
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629 poni ex rationibus temporum BE ad GI, &amp; ex ea veloci-
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630 tatem extremarum ED ad IK. Fiat vt BE ad GI, ita BC
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631 ad GH, intelligatur que GHLI figura $imilis ip$i BDE. Quo-
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632 <MARG><I>Pr.</I> 2. <I>huiu.</I></MARG>
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633 niam $patium A ad F, hoc e$t imago BCDE ad imaginem
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634 GMKI componitur ex ratione imaginis BCDE ad figu-
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635 ram $ibi $imilem GHLI, &amp; ex ratione huius ad imaginem
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636 GMKI: prior ratio e$t duplicata homologorum lateru&mtail;
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637 BE ad GI, $eu e$t compo$ita ex BE ad GI, &amp; ex huic $imi-
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638 li ratione ED ad IL, &amp; ratio altera, imaginis $cilicet GHLI
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639 ad imaginem GMKI e$t, vt LI ad IK; ergo ex &aelig;quali ima-
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640 go BCDE ad imaginem GMKI, hoc e$t $patium A ad $pa-
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641 tium F, componetur ex ratione temporum BE ad GI, &amp; ex
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642 rationibus ED ad LI, &amp; IL ad IK, $cilicet nectetur ex ra-
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643 tione BE ad GI, &amp; ED ad IK, qu&aelig; po$trema cum $it ratio
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644 velocitatum extremarum ED ad IK; con$tat, quod propo-
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645 $uimus, $patia $imilium motuum componi ex ratione tem-
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646 porum, &amp; ex ratione homologarum velocitatum, hoc e$t
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647 extremarum.
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648 <C><I>Corollarium.</I></C>
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649 <p><I>Si tempora fuerint &aelig;qualia, $imilium motuum $patia er&utilde;t
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650 vt extrem&aelig;, vel $umm&aelig; velocitates, &amp; contra, $i i$t&aelig; &aelig;quales
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651 $int, erunt $patia vt tempora.</I>
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652 <C><I>Corollarium II.</I></C>
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653 <p><I>Cum $patia $imilium motuum nect antur ex ratione tem-
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654 porum &amp; ex ea velocitatum $ummarum, $eu earum, qu&aelig; s&utilde;t
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655 ad in$tantia $imiliter $umpta in rectis BE, GI, const at ex
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656 lem: infra cor.</I> 2. <I>pr.</I> 3. <I>huius tempora componi ex rationi-
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657 bus $patiorum $imilium motuum, &amp; ex rec&igrave;proca dictarum</I>
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658 <pb n=18>
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659 <I>velocitatum. Ex eadem ratione patet e$$e velocitates $um-
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660 mas, vel homologas vti diximus in ratione compo$ita dicto-
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661 rum $patiorum, &amp; ip$orum temporum.</I>
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662 <C><I>Corollarium III.</I></C>
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663 <p><I>Quare $i alter&aelig; de du<*>bus componentibus &aelig;qualis fuerit,
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664 reliqua tant&ugrave;m computanda erit.</I>
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665 <C><I>Scholium.</I></C>
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666 <p><I>Hinc emergit <*>nis fer&egrave; doctrina grauium cum de$cend&utilde;t
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667 pror$us libera, aut $uper planis inclinatis ad horizonte&mtail;:
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668 nec accidit veritates iam patefact as huc rur$us lectoris tadio
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669 afferre, $ed libcat potius, rationem metiendarum imaginum,
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670 quamuis longitudine immen$arum, no$tra methodo exponere.</I>
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671 <C>DEF. VIII.</C>
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672 <MARG><I>Tab.</I> 2. <I>Fig.</I> 6.</MARG>
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673 <p>SInt inter binas parallelas AB, GH, et IK, PQ plan&aelig; fi-
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674 gur&aelig; ABHG, IKQP, &amp; in altera earum ducta altitudi-
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675 ne RV, $int inter $e ip$&aelig; figur&aelig; talis natur&aelig;, vt cum $it
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676 GABH ad $egmentum EABF factum per &aelig;quidi$tantem
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677 ip$i GH $icut VR ad RT, verificetur $emper (ducta &aelig;qui-
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678 di$tanti NTO ip$i PQ) e$$e GH ad EF vt reciproc&egrave; NO ad
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679 PQ. tunc huiu$modi figuras vocabimus inter $e auuer$as.
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680 <C><I>Corollarium.</I></C>
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681 <p><I>Sequitur ex vi nunc allat&aelig; deffi<*>, lineam IK tunc e$$e in-
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682 finitam, cum AB fuerit punctum, &amp; ideo $imul con$tat figu-
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683 ram IPQK immen$am e$$e longitudine vers&ugrave;s K aut I, aut
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684 vtrinque, $i nempe producerentur nunquam co<*>ur&aelig; line&aelig;
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685 QP, IK.</I>
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686 <pb n=19>
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687 <C>PROP. IX. THEOR. IX.</C>
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688 <p>REctangulum $ub altitudine, &amp; ba$i vnius auuer$arum
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689 ad ip$am auuer$am figuram, eandem habet ration&etilde;,
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690 ac altera auuer$a figura ad rectangulum ex ba$i in altitudi-
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691 <MARG><I>Tab.</I> <*>. <I>fig.</I> 7.</MARG>
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692 nem eiu$dem huius figur&aelig;.
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693 <p>Sint auuer$&aelig; figur&aelig; ACB, GFDEG. Dico rectangu-
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694 lum DF in DE ad figuram GFDEG, eandem habere ratio-
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695 nem ac figura ACBA ad rectangulum AB in BC. Sint pri-
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696 m&ugrave;m ABC, FDE anguli recti, &amp; ducta qualibet HI paral-
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697 <MARG><I>Def.</I> 8. <I>huius.</I></MARG>
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698 lela BC, $it BAC ad HIA vt DF ad KF, erit ob naturam
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699 auuer$arum KL ad DE vt BC ad HI; itaque $i ponatur e$$e
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700 quidam motus ab F in D iuxta imaginem velocitat&utilde; BAC,
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701 <MARG><I>Def:</I> 2. <I>huius.</I></MARG>
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702 erit GFDEG imago temporis eiu$dem motus; nam imago
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703 <MARG><I>pr.</I> 2. <I>huius.</I></MARG>
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704 BAC ad imaginem HIA e$t vt $patium DF ad $patium FK
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705 &amp; velocitas BC ad velocitat&etilde; HI vt reciproc&egrave; KL ad DE.
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706 Sit etiam alius motus, $ed &aelig;quabilis, cuius imago velocita-
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707 tum &aelig;qualis $it, &amp; homogenea ip$i BAC, rectangulum n&etilde;-
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708 pe AB in BM, &amp; ideo $i fiat BM ad BC $icut DE ad DN,
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709 concipiaturque rectangulum FD in DN, erit hoc imago
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710 <MARG><I>Def.</I> 2. <I>huius.</I></MARG>
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711 temporis dicti motus &aelig;quabilis, homogenea, &amp; &aelig;qualis
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712 imagini GFDEG; nam t&etilde;pora, $cilicet imagines GFDEG,
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713 <MARG><I>pr.</I> 1. <I>huius.</I></MARG>
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714 FD in DN rectangulum componuntur ex rationibus $pa-
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715 <MARG><I>pr.</I> 4. <I>huius.</I></MARG>
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716 tiorum, hoc e$t imaginum velocitatum inter$e &aelig;qualium,
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717 ABM, ACB, &amp; reciproca &aelig;quatricum pariter &aelig;qualium
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718 BM, BM. Cum igitur rectangulum FD in DN &aelig;quale $it
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719 <MARG><I>Cor. pr.</I> 3. <I>hu-
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720 ius.</I></MARG>
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721 imagini, $eu figur&aelig; GFDEG, habebit eadem figur&atail;
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722 GFDEG ad rectangulum FD in DE eandem rationem,
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723 quam DN ad DE, hoc e$t quam BC ad BM, $eu quam re-
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724 ctangulum AB in BC ad rectangulum AB in BM, aut ad ei
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725 &aelig;qualem figuram ABC; &amp; conuertendo, manife$tum e$t
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726 quod propo$uimus, nempe rectangulum FD in DE ad fi-
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727 guram GFDEG habere eandem ration&etilde;, ac figura ACBA
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728 <pb n=20>
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729 ad rectangulum AB in BC. quod erat demon$trandum
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730 primo loco.
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731 <p>2. Si ver&ograve; propo$it&aelig; figur&aelig; $int qu&aelig;cunque auuer$&aelig;
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732 <MARG><I>Tab.</I> 2. <I>Fig.</I> 8.</MARG>
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diff changeset
733 DAE, QPLMQ poterunt h&aelig; reuocari ad qua$dam alias
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diff changeset
734 FKG, RSZX, qu&aelig; $int inter ea$dem parallelas, queis com-
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diff changeset
735 prehenduntur propo$it&aelig; figur&aelig;, ad eo vt exi$tentibus re-
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diff changeset
736 ctis angulis KFG, RXZ $int ip$&aelig; bin&aelig; figur&aelig; ab ij$dem pa-
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diff changeset
737 rallelis intercept&aelig;. inter $e &aelig;qualiter analog&aelig; hoc e$t du-
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diff changeset
738 ctis &aelig;quidi$tantibus, vt vi$um fuerit IHBC, VTNO, $int
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parents:
diff changeset
739 $emper interiect&aelig; line&aelig; IH, BC, &amp; VT, NO &aelig;quales: hoc
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diff changeset
740 modo non tant&ugrave;m liquet figuras FKG, DAE, nec no&ntail;
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diff changeset
741 RSZX, PQML &aelig;quales inter $e e$$e, ver&ugrave;m etiam FKG ad
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diff changeset
742 IKH e$$e in eadem ratione, in qua QPLMQ ad QPNOQ,
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diff changeset
743 quamobrem ex prima parte, rectangulum ZX in RM ad
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diff changeset
744 figuram SRXZS, hoc e$t rectangulum LM in altitudinem
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diff changeset
745 figur&aelig; QPLMQ ad hanc ip$am figuram habebit eandem
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diff changeset
746 rationem, quam figura FKG ad rectangulum KF in FG,
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diff changeset
747 vel quam figura DAE ad rectangulum DE in altitudinem
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diff changeset
748 eiu$dem huius figur&aelig; DAE; quo circa con$tat omne pro-
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diff changeset
749 po$itum.
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parents:
diff changeset
750 <C><I>Corollarium.</I></C>
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diff changeset
751 <p><I>Patet in prima parte repertum e$$e rectangulum FD i&ntail;</I>
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diff changeset
752 <MARG><I>Cor. pr.</I> 18.
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diff changeset
753 <I>huius.</I></MARG>
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diff changeset
754 <I>DN &aelig;quale figur&aelig; GFDEG, lic&egrave;t h&aelig;c immen$e longitudinis
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diff changeset
755 $it vers&ugrave;s G, &amp; ob id manife$tum e$t, qu&ograve;d quamuis aliqu&atail;
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diff changeset
756 figura $it $in&egrave; fiue longa, non ideo $emper magnitudinem ha-
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diff changeset
757 bet infinitam. Et $imul illud con$tat, vbi vna auuer$arum, $eu
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diff changeset
758 vbi imago velocitatum, aut temporis $it magnitudine termi-
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diff changeset
759 nata, etiam altera auuer$arum, vel imaginum erit huiu$-
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diff changeset
760 modi &amp;c.</I>
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diff changeset
761 <pb n=21>
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diff changeset
762 <C>PROP. X. THEOR. X.</C>
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763 <p>IN quouis parallelogrammo BD $int deinceps diagona-
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diff changeset
764 <MARG><I>Tab.</I> 2. <I>Fig.</I> 9.</MARG>
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diff changeset
765 les AGC, AHC, AIC, ALC, ali&aelig;que numer&ograve; infinit&aelig;,
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diff changeset
766 ita vt acta qu&aelig;libet recta EF parallela BA $ec &atilde;s ip$as dia-
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diff changeset
767 gonales in punctis G, L, H, I, $it $emper DA ad AF, vt CD,
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768 aut EF ad FG; quadratum ex DA ad quadratum AF vt
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diff changeset
769 EF ad FH; cubus ex DA ad cubum ex AF vt EF ad FI;
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diff changeset
770 quadroquadratum ex DA ad quadroquadratum ex AF
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771 vt EF ad FL; &amp; $ic continu&ograve; procedendo per infinitas ex
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diff changeset
772 ordine pote$tates: Stephanus de Angelis Author $ubtilis,
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773 ac celeberrimus, libro $uo infin. parabolarum vocat trian-
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diff changeset
774 gulum rectilineum ABC parabolam primam, BAHC $e-
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diff changeset
775 cundam; tertiam BAIC, quartam BALC, &amp; ita in infini-
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diff changeset
776 tum: His definitis docet ex Cauallerio parallelogrammum
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777 BD ad quancunque dictarum parabolarum $ibi in$cripta-
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778 rum e$$e vt numerus, vel exponens parabol&aelig; vnitate au-
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779 ctus ad ip$um exponentem, $iue numerum parabol&ecedil;, qua-
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diff changeset
780 re ad primam habebit ip$um parallelogrammum eandem
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diff changeset
781 rationem, ac 2 ad 1; ad $ecundam vt 3 ad 2; ad tertiam vt
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782 4 ad 3, &amp; ita deinceps de reliquis; itaque per conuer$io-
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783 nem rationis habebit ip$um parallelogrammum ad exce$-
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784 $um illius $upra quancunque parabolarum dictarum, $cili-
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785 cet ad trilineum primum AGCD eandem rationem, quam
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786 2 ad 1, ad $ecundum quam 3 ad 1, &amp; $ic deinceps quam
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787 numerus trilinei vnitate auctus ad ip$am vnitatem. Sed
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diff changeset
788 e$t etiam admonendum verticem dictarum parabolarum
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789 e$$e punctum A, &amp; per con$equens AB diametrum, &amp; BC
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790 ordinatim aplicatam, $eu ba$im.
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791 <pb n=22>
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792 <C>PROP. XI. THEOR. XI.</C>
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793 <p>II$dem adhuc manentibus, idem de Angelis mon$trat eo-
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794 dem illo tractatu pr. 3. $i qu&aelig;cunque ex dictis parabo-
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795 lis $ecta $it qualibet recta parallela ba$i BC, e$$e parabolam
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796 ad re$ectam portionem ver$us verticem, vt pote$tas ba$is,
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797 cuius exponens e$t numerus parabol&aelig; vnitate auctus ad
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798 $imilem pote$tatem ex ba$i re$ect&aelig; portionis; itaque i&ntail;
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799 prima parabola e$t vt quadratum ad quadratum, in $ecun-
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parents:
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800 da vt cubus ad cubum, &amp; $ic de c&aelig;teris. Similiter $i $ece-
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801 tur quodlibet ex infinitis trilineis linea GF ba$i CD paral-
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802 lela, erit trilineum ad $uperius $ui $egmentum vt pote$tas
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803 ex DA, cuius exponens e$t numerus trilinei vnitate auctus
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804 ad $imilem pote$tatem ex AF. quare trilineum primu&mtail;
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805 CAD ad GAF erit vt quadratum ex DA ad quadratum
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806 ex FA, $ecundum CHAD ad $egmentum HAF vt cubus
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807 ad cubum, &amp; ita in c&aelig;teris eodem ordine.
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808 <C>PROP. XII. THEOR. XII.</C>
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809 <MARG><I>Tab.</I> 3. <I>fig.</I> 1.</MARG>
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810 <p>SIt mod&ograve; ACD angulus rectus, &amp; linea FE talis natur&aelig;,
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811 vt deductis ad libitum rectis AF, BE parallelis ip$i
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812 CD, pote$tas ex CA ad $imilem pote$tatem ex CB $it reci-
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813 proc&egrave; vt alia qu&aelig;dam pote$tas ex BE ad $imilem huic po-
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814 te$tatem ex AF; patet rectas CA, CD nondum iungi cum
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815 EF, quamuis in immen$um vn&agrave; producerentur. Ab hoc
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816 proprietate VValli$ius &amp; Fermatius $ubtili$$imi authores
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817 vocauerunt curuam FE nouam hyperbolam, &amp; eius a$-
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818 $ymptotos AC, CD. Omnes huiu$modi hyperbol&aelig;, qu&aelig;
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819 infinit&aelig; numero $unt, terminantur ad vnam partem ma-
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820 gnitudine, cum hyperbola c&otilde;munis, $eu Apolloniaca $it in
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821 vtranque partem magnitudine infinita. Quod ergo exi-
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822 mium e$t, o$tenderunt ip$i authores rectangulum FA i&ntail;
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823 <pb n=23>
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824 AC ad $patium hyperbolicum qu&agrave; finitum e$t, lic&egrave;t $in&egrave;
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825 fine longum, eandem habere rationem, quam differentia
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826 exponentium pote$tatum hyperbol&aelig; ad exponentem po-
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827 te$tatis minoris. Quare $i in hyperbola $it vt cubus CB
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828 ad cubum CA ita quadratum AF ad quadratum BE, erit
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829 pr&aelig;dictum rectangulum CA in AF dimidium Spatij $in&egrave;
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830 fine producti A &amp; FA; at $i quadratum CB ad quadratum
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831 CA $it vt recta AF ad rectam BE, rectangulum ip$um CA
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832 in AF &aelig;quale erit $patio A &amp; FA, qu&ograve;d $i pote$tas CA vel
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833 CB non fuerit altior pote$tate ex BE, vel AF, tunc ip$um
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834 illud $patium, infinitum quoque erit magnitudine, etenim
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835 nullus exce$$us exponentis pr&aelig;dict&aelig; pote$tatis ex CA $u-
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836 pra exponentem pote$tatis BE, habet ad numerum expo-
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837 nentis pote$tatis BE rationem infinitam.
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838 <C>DEMONSTRATIO.</C>
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839 <p>SVpradictum propo$itum habetur in commercio epi-
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840 $tolico Ioannis Valli$ij Epi$tola quarta, quem libellum
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841 vn&agrave; cum alijs docti$$imis $uis operibus Vincentius Viuia-
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842 nus ingens &aelig;ui no$tri Geometra, antequam $umma cu&mtail;
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843 humanitate mi$i$$et, eidem ip$i quadraturam vnius ex di-
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844 ctis hyperbolis ex no$tris principijs deductam, ac excogi-
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845 tatam, indicauimus. Cum ver&ograve; po$tea nobis eueni$$et
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846 vniuer$aliorem ad alias hyperbolas ($emper communi ex-
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847 cepta) accomodatam reperij$$e, huc debemus afferre, pri-
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848 m&ugrave;m vt quendam fructum $cienti&aelig; huius; deinde cum di-
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849 ctorum authorum ip$am propo$itionis demon$trationem
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850 non habuerimus, &amp; demum quia ip$arum hyperbolarum
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851 men$ura, ac quadratura in aquarum rationibus erunt po-
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852 ti$$imum ex v$u. Sit igitur BC vna ex infinitis hyperbolis,
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853 <MARG><I>Tab.</I> 3. <I>Fig.</I> 3.</MARG>
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854 quarum a$lymptoti AE, EL; Sint etiam qu&aelig;cunque apli-
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855 cat&aelig; AB, DC alsymptoto EL &aelig;quidi$tantes, &amp; habeat
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856 DE ad EA eandem rationem v. g. quam cubus ex AB ad
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857 <pb n=24>
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858 <MARG><I>Tab.</I> 3. <I>fig</I> 2.</MARG>
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859 cubum DC. Patet $i proponeretur illi auuer$a figur&atail;
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860 <MARG><I>Def.</I> 8. <I>huius.</I></MARG>
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861 FGK, e$$etque AE ad DE vt figura GFK ad figuram IHK
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862 e$$e etiam FG ad IH vt DC ad AB, e$t autem cubus ex
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863 DC ad cubum ex AB vt AE ad ED; ergo etiam figur&atail;
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864 FGK ad IHK ($unt enim FG, IH parallel&ecedil;) habebit ean-
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865 dem rationem, ac cubus ex FG ad cubum ex IH: Itaqu&etail;
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866 GFK erit comunis parabola, hoc e$t quadratica, $eu $ec&utilde;-
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867 <MARG><I>Pr.</I> 10. <I>huius.</I></MARG>
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868 da in $erie infinitarum parabolarum, &amp; ob id eadem GFK
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869 <MARG><I>Pr.</I> 9. <I>huius.</I></MARG>
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870 parabola ad rectangulum GF in FK erit vt 2 ad 3, in qua
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871 ratione $e habebit quoque rectangulum BA in AE ad $pa-
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872 tium infinit&egrave; longum &amp; BM, et erit vt 2 ad 1; $cilicet vt ex-
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873 ce$$us exponentis maioris pote$tatis, qu&aelig; cubica e$t, $uper
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874 numerum exponentis, qui hoc ca$u e$t tant&ugrave;m vnitas ra-
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875 dicis, e$t ad hunc ip$um exponentem, $eu vnitatem line&aelig;
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876 indicantem, quod concordat cum propo$ita dictoru&mtail;
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877 authorum.
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878 <C><I>Exemplum aliud.</I></C>
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879 <MARG><I>In eadem fi-
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880 gur&aelig;.</I></MARG>
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881 <p>SIt etiam cubus ex DE ad cubum ex AE, $icut quadra-
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882 to quadratum AB ad quadroquadratum DC, &amp; rur-
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883 $us propo$ita GKF auer$a huius hyperbol&aelig;: patet $i $it AE
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884 ad DE vt figura GFK ad figuram IKH, e$$e etiam FG ad
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885 <MARG><I>Def.</I> 8. <I>huius.</I></MARG>
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886 IH vt DC ad AB; cumque $it cubus ex AE ad cubum ex
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887 DE $icut quadroquadratum ex DC ad quadroquadrat&utilde;
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888 ex AB, erit etiam quadroquadratum ex FG ad quadro-
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889 quadratum ex IH, vt cubus ex AE ad cubum ex DE; $i
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890 igitur intelligatur qu&aelig;dam ratio, qu&aelig; $it $ubduodecupla
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891 tam rationis quadroquadratorum qu&agrave;m huic $imilis cu-
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892 borum pr&aelig;dictorum, erit porr&ograve; FG ad IH triplicata, &amp;
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893 AE ad ED quadruplicata eiu$dem dict&aelig; $ubduodecupl&aelig;;
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894 quamobrem etiam ratio figur&aelig; GFK ad figur&atilde; IHK, qu&aelig;
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895 e$$e debet vt AE ad ED, erit quadruplicata eiu$dem $ub-
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896 duodecupl&aelig;: &amp; ide&ograve; $i ponamus IK ad KI in ratione
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897 <pb n=25>
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898 eiu$dem $ubduodecupl&aelig;, erit figura GFK illius natur&aelig;, vt
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899 <MARG><I>Pr.</I> 10. <I>huius.</I></MARG>
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900 $it $emper cubus ex FK ad cubum ex KI $icut GF ad IH, &amp;
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901 hoc modo eadem illa figura erit trilineum tertium, $eu cu-
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902 bicum, ex quo ergo $equitur, GFK ad HIK $it in eadem ra-
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903 tione, in qua quadroquadratum ex FK ad quadroqua-
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904 dratum ex KI, hoc e$t $it vt AE ad ED; $equiturque etiam
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905 <MARG><I>Pr.</I> 10. <I>huius.</I></MARG>
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906 ob hoc figuram GFK $ubquadruplam e$le circum$cripti
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907 rectanguli GF in FK; e$t autem vt trilineum GFK ad rect&atilde;-
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908 <MARG><I>Pr.</I> 9. <I>huius</I></MARG>
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909 gulum GF in FK circum$criptum, $ic rectangulum ABME
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910 ad auuer$am eidem trilineo figuram AB &amp; EA, ergo re-
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911 ctangulum ABME $ubquadruplum erit eiu$dem figur&aelig;
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912 AB &amp; EA longitudinis infinit&aelig;, quare ip$um rectangulum
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913 erit $ubtriplum portionis &amp; BM &amp; longitudinis pariter im-
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diff changeset
914 men$&aelig;. Cum ita $it, con$tat exemplo hoc quoque, eand&etilde;
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diff changeset
915 illam rationem e$$e exce$$um maioris exponentis $upr&atail;
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diff changeset
916 minorem exponentem ad hoc ip$um, dictarum pote$tat&utilde;
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diff changeset
917 hyperbol&aelig;.
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diff changeset
918 <C>PROP. XIII. THEOR. XIII.</C>
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diff changeset
919 <p>SVperior demon$tratio effecta fui$$et ampli$$ima, $i pr&ecedil;-
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diff changeset
920 ponere volui$$emus quadratur&atilde; vt datam omnis ge-
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diff changeset
921 neris parabolarum, &amp; trilineorum, ver&ugrave;m cum i$ta pars n&otilde;
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diff changeset
922 $it plen&egrave; tradita, vt videre e$t quinto libro infinitarum pa-
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diff changeset
923 rabolarum eiu$dem de Angelis, $atius ideo duximus qua-
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diff changeset
924 draturam hyperbolarum &agrave; VVali$io, &amp; Fermatio acuti$$i-
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diff changeset
925 mis illis viris propo$itam omnino veram admittere, vt ind&egrave;
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diff changeset
926 eam parabolarum &amp; trilineorumvniuer$alem, quam adhuc
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927 ab alijs non habemus, facillim&egrave;, compendios&egrave;que depro-
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diff changeset
928 meremus. Hanc igitur ita proponimus vt $ubinde o$ten-
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diff changeset
929 damus.
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diff changeset
930 <p>Si $imiles pote$tates applicatarum fuerint in eadem ra-
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diff changeset
931 tione, ac $unt inter$e pote$tates qu&aelig;dam ali&aelig;, &amp; eiu$dem
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diff changeset
932 gradus diametrorum ab ip$is applicatis ab$ci$$arum v$que
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diff changeset
933 <pb n=26>
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934 ad verticem parabolarum, vel trilineorum; erit rectangu-
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diff changeset
935 lum ad parabolam $ibi in$criptam vt aggregatum expon&etilde;-
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diff changeset
936 tium vtriu$que pote$tatis ad exponentem altioris ip$arum
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diff changeset
937 pote$tatum parabol&aelig;; &amp; ad trilineum vt aggregatum ex-
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diff changeset
938 ponentium pote$tatum trilinei ad exponentem inferioris
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diff changeset
939 pote$tatis eiu$demmet trilinei. Sic enim in expo$ita figu-
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940 ra pr&aelig;dicta, $i e$$et quadratum ex FG ad quadratum ex
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diff changeset
941 IH, $icut cubus ex FK ad cubum ex IH, e$$et rectangulum
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diff changeset
942 GF in FK ad figuram GFK (qu&aelig; tunc foret trilineum, vt
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diff changeset
943 5 ad 2; nam vbi pote$tas ab$ci$$arum maior e$t illa applica.
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diff changeset
944 tarum e$t $emper GF trilineum. Simili modo, $i $it vt qua-
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diff changeset
945 dratum ex FK ad quadratum ex KI ita cubocubus ex FG
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diff changeset
946 ad cubocubum ex IH; hoc e$t $i $it cubus ex FG ad cub&utilde;
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diff changeset
947 ex IH, vt linea FK ad KI (tolluntur enim vtrinque ex $imi-
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diff changeset
948 libus $imiles rationes) erit $igura GFK parabola, ad quam
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diff changeset
949 $ibi circum$criptum rectangulum eandem habebit ration&etilde;,
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diff changeset
950 quam 4 ad 3, &amp; $ic dicendum erit de omnibus alijs para-
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951 bolis atque trilineis.
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diff changeset
952 <C>DEMONSTRATIO.</C>
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953 <p>VEr&ugrave;m vt propo$itum o$tendamus, e$to qu&aelig;libet ex
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diff changeset
954 parabolis GFK, nimirum quadratocubus ex FG ad
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955 quadratocubum ex IH habeat eandem rationem, qua&mtail;
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956 cubus ex FK ad cubum ex IK. Demon$tro, rectangulum
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957 GF in FK habere eandem rationem ad parabolam GFK,
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958 quam aggregatum exponentium 8 ad maiorem exponen-
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959 tem 5. Prim&ugrave;m, quam rationem habet rectangulum GF in
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960 FK ad parabolam GFK, eandem habebit rectangulum HI
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961 in IK ad parabolam HIK (hoc enim demon$trabimus in-
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962 fr&agrave;) permutandoque, erit rectangulum GF in FK ad re-
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diff changeset
963 ctangulum HI in IK, vt parabola GFK ad parabolam HIK;
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964 componuntur ver&ograve; illa rectangula ex rationibus GF ad
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965 IH, &amp; FK ad IK, ergo etiam parabola ad parabolam com-
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966 <pb n=27>
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967 ponetur ex ij$dem rationibus; &amp; quoniam ductis inuicem
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968 exponentibus po$$unt con$iderari quindecim rationes in-
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969 ter $e $imiles, ex quibus con$tet tam ratio dictorum cubo-
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diff changeset
970 rum, qu&agrave;m huic $imilis altera quadratocuborum, &amp; tunc
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971 GF ad IH erit triplicata, et FK ad KI quintuplicata eiu$d&etilde;
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972 $ubquindecupl&aelig; rationis, qu&aelig; $it A ad B; ergo $imul ad-
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973 ditis ij$dem rationibus, quintuplicata $cilicet, &amp; triplicata
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974 exiliet ratio octuplicata ip$ius A ad B; proptereaque pa-
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975 rabola GFK ad HIK, $eu $i con$ideremus figuram &amp; BAEL
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976 auuer$am parabol&aelig; GFK, ita vt AE ad ED $it vt para-
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977 <MARG><I>Def.</I> 8. <I>huius.</I></MARG>
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978 bola GFK ad parabol&atilde; HIK; AE ad ED erit pariter octu-
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979 plicata eiu$dem A ad B; &amp; cum $it ob naturam auuer$ar&utilde;
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980 FG ad HI vt DC ad AB; erit DC ad AB triplicata eiu$d&etilde;
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981 rationis A ad B, qnare vt cubus AE ad cubum DE, it&atail;
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982 quadratocubocubus DC ad quadratocubocubum ex
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983 AB: rectangulum igitur ABME ad $patium hyperbolicum
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984 infin<*>&egrave; longum &amp; BM &amp; erit vt quinque ad tria, &amp; ad vni-
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985 <MARG><I>Pr.</I> 12 <I>huius.</I></MARG>
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986 uer$um $pa ium &amp; BAE &amp; vt 5 ad 8, in qua nempe ratio-
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987 ne debet e$$e parabola GF<I>K</I> ad rectangulum GF in FK.
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988 <MARG><I>Pr.</I> 9. <I>huius.</I></MARG>
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989 Quod &amp;c.
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990 <C><I>Corollarium.</I></C>
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991 <p><I>Con$tat $i fuerit ratio A ad B e&ograve; $ubmultiplicata rationis
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992 applicatarum, quoties e$t numerus exponentis pote$tatis ab-
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993 $ci$$arum eiu$dem parabol&aelig;, e$$e ip$am parabolam ad $ui por-
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994 tionem in tam multiplicata ratione A ad B, ac e$t numerus
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995 aggregati exponentium ambarum pote$tatum parabola. Nam
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996 cum e$$et quadratocubus ex FG ad quadratocubum ex IH, $i-
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997 cut cubus ex FK ad cubum ex IK, propo$ita in$uper e$$et A ad
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998 B. $ubquindecupla alterius dictarum $imilium rationum ex
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999 pote$t atibus parabola, o$ten$um fuit rationem A ad B $ubtri-
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1000 plicatam ip$ius GF ad IH, &amp; $ubquintuplicatam alterius FK
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1001 ad KI, &amp; tandem o$tendimus parabolam GFK ad portionem</I>
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1002 <pb n=28>
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1003 <I>eius HIK e&szlig;e in octuplicata ratione eiu$dem A ad B; quod
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1004 idem omnino diceretur $i figura GFK trilineum e$$et. Ratio
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1005 autem A ad B dicetur impo$terum logarithmica pote$tatum
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1006 parabol&aelig;, $eu trilinei, aut hyperbol&aelig;.</I>
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1007 <C>ASSVMPTVM.</C>
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1008 <p>REliquum e$t vt o$tendamus, parabolam GFK ad
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1009 portionem HIK e$$e vt rectangulum GF ad rectan-
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1010 gulum HI in IK, $cilicet e$$e in ratione compo$ita ha$ium,
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1011 &amp; altitudinum parabolarum, quod nempe $ic o$tendetur,
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1012 Sit vt $upra FGK parabola, eiu$que portio IHK; exi$tenti-
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1013 bus ver&ograve; applicatis FG, IH, fiat EG ad IE vt FK ad KI, $it-
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1014 <MARG><I>Tab.</I> 3. <I>Fig.</I> 2.</MARG>
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1015 que IE ba$is, et K vertex parabol&ecedil; IEK $imilis ip$i GFK pa-
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1016 tet propter $imilitudinem figurarum, e$$e parabolam GFK
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1017 ad parabolam IEK in eadem duplicata ratione FG ad IE,
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1018 in qua nempe e$t rectangulum GF in FK ad $ibi $imile re-
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1019 ctangulum EI in IK, ob idque rectangulum GF in FK ad
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1020 rectangulum EI in IK, cum $int inter$e vt parabola GFK ad
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1021 parabolam EIK, h&aelig;c ver&ograve; parabola ad ip$am IHK habeat
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1022 eandem rationem, ac IE ad IH; $eu ob eandem altitudinem
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1023 IK vt rectangulum EI in IK ad rectangulum HI in IK, erit
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1024 ex &aelig;quali parabola GFK ad parabolam HIK vt rectangu-
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1025 lum GF in FK ad rectangulum HI in IK. Quod &amp;c.
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1026 <C>PROP. XIV. THEOR. XIV.</C>
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1027 <MARG><I>Tab.</I> 2. <I>fig.</I> 3.</MARG>
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1028 <p>IN quacunque hyperbola (excepta $emper conica) cu-
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1029 ius a$$ymptoti EA, EM, $i $it pote$tas applicatarum DC
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1030 AB altior pote$tate ab$ci$$arum AE, ED ($ic enim finit&atail;
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1031 erit magnitudine $ecundum eam a$$ymptoton, qu&aelig; appli-
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1032 catis parallela e$t) $patium ip$um hyperbol&aelig; &amp; BAE &amp;
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1033 ad $ui portionem &amp; CDE &amp; habebit eandem rationem, ac
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1034 rectangulum BAE ad rectangulum CDE, $eu (a$$umpta
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1035 <pb n=29>
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1036 ratione logarithmica A ad B pote$tatum hyperbol&aelig;) qu&atilde;
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1037 pote$tas ex A, cuius exponens e$t differentia exponenti&utilde;
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1038 pote$tatum hyperbol&aelig; ad $imilem pote$tatem ex B.
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1039 <C>DEMONSTRATIO.</C>
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1040 <p>QVam rationem habet rectangulum BAE ad $patium
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1041 &amp; BAE &amp;, eandem habet rectangulum CDE ad
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1042 <MARG><I>Pr.</I> 12. <I>huius.</I></MARG>
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1043 $patium &amp; CDE, &amp; permutando erit rectangu-
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1044 lum BAE ad CDE, $icut $patium &amp; BAE &amp; ad $patiu&mtail;
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1045 &amp; CDE &amp;; $i igitur in eadem propo$ita hyperbola $it po-
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1046 te$tas applicatarum DC, AB quintuplicata ip$ius A ad B,
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1047 &amp; AE ad ED $eptuplicata $it eiu$dem; erit $eptuplicat&atail;
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1048 applicatarum in eadem ratione, ac quintuplicata ab$ci$$a-
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1049 rum; $cilicet quadratoquadratocubus ex DC ad $imilem
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1050 pote$tatem ex AB erit vt quadratocubus ex AE ad qua-
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1051 dratocubum ex DE, eritque $ic maior pote$tas applicata-
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1052 rum, atque adeo componetur rectangulum EAB ad EDC
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1053 ex $eptuplicata ip$ius A ad B, qualis e$t AE ad ED, &amp; $ub-
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1054 quintuplicata eiu$dem A ad B, qu&aelig; e$t AB ad DC; nimi-
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1055 r&ugrave;m erit rectangulum EAB ad EDC in duplicata tantum
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1056 ratione ip$ius A ad B: quare $patium &amp; BAE &amp; ad id
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1057 &amp; CDE &amp;, qu&aelig; $unt inter $e, vt ip$a rectangula, erit vt po-
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1058 te$tas ex A, cuius exponens e$t differentia exponentium &amp;
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1059 S pote$tatum hyperbol&aelig; ad $imilem pote$tatem ex B.
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1060 Quod &amp;c.
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1061 <C>PROP. XV. THEOR. XV.</C>
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1062 <p>SIab exponente pote$tatis applicatarum hyperbol&ecedil; de-
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1063 trahatur exponens minoris pote$tatis ab$ci$$arum, po-
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1064 te$tas reliqui exponetis erit applicatarum auuer$&aelig; figur&aelig;,
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1065 in ab$ci$$is ver&ograve; ade$t vtrobique eadem pote$tas. Itaque
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1066 cum in $uperiori hyperbola re$idui exponentis pote$tas
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1067 <pb n=30>
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1068 quadratum e$$et, porr&ograve; in eius auuer$a e$$et pote$tas appli-
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1069 catarum quadratica, &amp; ab$ci$$arum quadratocubica.
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1070 <C>DEMONSTRATIO.</C>
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1071 <MARG><I>Tab.</I> 3. <I>Fig.</I> 3.</MARG>
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1072 <p>ESto rur$us hyperbola &amp; BAE &amp;, et $icut dictum e$t
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1073 AE ad ED $it in $eptuplicata ratione logarithmic&aelig;
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1074 rationis A ad B, at DC ad AB in quintuplicata, videlicet
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1075 quadratocubus ex AE ad quadratocubum ex DE eandem
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1076 habeat rationem, ac quadratoquadratocubus ex DC ad
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1077 $imilem pote$tatem ex AB; Dico in auuer$a figura pote$ta-
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1078 tem aplicatarum e$$e quadratum, cuius expon&etilde;s 2 e$t dif-
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1079 ferentia exponentium pote$tatum hyperbol&aelig;; pote$tatem
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1080 ver&ograve; ab$ci$$arum eandem e$$e, ab$ci$$arum eiu$dem hyper-
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1081 bol&aelig;. Sit vt $upra FK ad KI vt hyperbola &amp; BAE &amp; ad
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1082 &amp; CDE &amp;, hoc e$t, $it vt pote$tas ex A, cuius exponens
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1083 <MARG><I>Pr.</I> 14. <I>huius.</I></MARG>
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1084 e$t differentia exponentium pote$tatum hyperbol&aelig; ad $i-
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1085 milem pote$tatem ex B, &amp; ideo FK ad KI erit duplicata ip-
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1086 $ius A ad B, $ed DC ad AB eiu$dem illius logarithmic&aelig;
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1087 quintuplicata; e$tque in hac eadem ratione etiam GF ad
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1088 IH; ergo cum duplicata huius $it $imilis quintuplicat&aelig; KF
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1089 ad KI (nam vtraque ratio continet decies A ad B) pater,
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1090 quadratum ex FG ad quadratum ex IH e$$e eam pote$ta-
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1091 tem, quam propo$uimus euenire in applicatis auuer$&aelig;, cum
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1092 ali&agrave;s in ab$ci$$is $it vtrobique pote$tas eadem, nempe qua-
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1093 dratocubi. Quod &amp;c.
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1094 <C><I>Corollarium.</I></C>
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1095 <p><I>Patet ex noto trilineo, vel parabola FGK e$$e in auuer$a,
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1096 $cilicet in hyperbola &amp; BAE &amp; (qu&aelig; tunc e$t $emper magnitu-
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1097 dine finita iuxta a$symptoton EM &amp;) pote$tatem applicatar&utilde;,
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1098 qua pro exponente habet $ummam exponentium pote$tatum
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1099 parabol&aelig;, aut trilinei; nam cum e&szlig;et in trilineo pracedenti</I>
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1100 <pb n=31>
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1101 <I>quadratum ex FG ad quadratum ex IH vt quadratocubus
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1102 ex FK ad quadratocubum ex IK, fuit equidem in hyperbol&atail;
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1103 quadratoquadratocubus ex DC<*> quadratoquadratocubum
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1104 ex AB $icut quadratocubus ex AE ad $imilem pote$tatem ex
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1105 DE, $cilicet inuariata pote$tate ab$cr$arum in ambabus au-
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1106 uer$is. Quare ex pote$tatibus notis vnius auuer$arum fa-
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1107 cil&egrave; inote$cent pote$tates alterius, atque etiam illius magnitu-
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1108 do. Nunc redeamus ad motus, nouamque adhuc methodum,
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1109 quam hoc loco re$eruauimus, afferamus.</I>
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1110 <C>DEF. IX.</C>
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1111 <p>SIt qu&aelig;dam Gene$is ACBH, cuius imago temporis
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1112 &amp; DCB &amp;; item $it FCBK gene$is alterius motus ab
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1113 <MARG><I>Tab.</I> 3. <I>fig.</I> 4.</MARG>
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1114 eodem C in B; &amp; act&agrave; rect&agrave; OIGE ip$i AFCD parallel&atail;,
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1115 ponantur CD, GE loco minimorum temporum, ita vt t&etilde;-
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1116 pore CD, dum mobile ex C affectum velocitate CA,
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1117 currat minimum $patiolum indicatum per C, cui e$t &aelig;qua-
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1118 le $patiolum aliud indicatum per G, quodque tran$igitur
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1119 tempore GE velocitate GD (nam vt e$$ent illa $patia i&ntail;
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1120 C, G &aelig;qualia, effectum fuit vt velocitas AC ad GD ean-
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1121 dem reciproc&egrave; rationem haberet, ac tempus GE ad CD,
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1122 id quod patet ex natura gene$is ACBH, &amp; imaginis &amp;
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1123 DCB &amp;) et hic rur$us notatu digni$$imum e$t nulli errori
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1124 obnoxium e$$e, qu&ograve;d &aelig;quabiles in illis minimis $patiolis
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1125 intellexerimus motus, quamuis potius deberet videri, in
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1126 ij$dem interuallis reperiri innumeras, ac in&aelig;quales veloci-
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1127 tates, queis nempe efficerentur motus in&aelig;quabiles, qu&ograve;d
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1128 gene$es in&aelig;quabiles $int. Cur i$ta $e ita habeant, hic non
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1129 e$t nobis di$putandum, ego enim puto, non ex indiui$ibili
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1130 velocitates alijs $uccedere, $ed reuera minutulum tempo-
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1131 ris con$iderari debere antequam motus diuer$imod&egrave; pro-
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1132 cedat, nempe ac $i velocitas, qu&aelig; $uccedere debet priori,
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1133 non ita $it in promptu, aut non ita $tatim mobile afficiat ad
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1134 <pb n=32>
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1135 motum $ibi proportionatum. Sed linquamus h&aelig;c alijs di$-
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1136 putanda: $atis nobis $it, methodum no$tram, quoad no$tr&utilde;
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1137 e$t, demon$trare. Ijs igitur vt $upra propo$itis, concipia-
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1138 tur adhuc tempore CD velocitate FC $pati&utilde; exigi quod-
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1139 dam, item aliud tempore EG, velocitateque GI, &amp; $ic per
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1140 omnes qua$cunque applicatas: qu&aelig;ritur, quod $patiu&mtail;
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1141 vltim&ograve; exactum e$$et, hoc e$t quam rationem id haberet ad
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1142 illud alterum $patium, quod eodem tempore tran$igitur
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1143 iu ta gene$im HACB, cuius imago temporis CD &amp; B.
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1144 I$ti duo motus in exemplo e$$ent, $i in quodam plano mo-
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1145 ueretur formica, dum ip$um planum vna eius extremitate
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1146 immobili circumduceretur, Sic formica difficili&ugrave;s a$c&etilde;de-
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1147 ret prout ip$um planum magis ad horizontem erigeretur.
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1148 Iam motus extremitatis plani circumact&aelig; habet gene$im
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1149 ACBH, cuius temporis imago &amp; DCB &amp;, et altera gene$is
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1150 FCBK <*>ribueretur motui formic&aelig;, nam vt dict&utilde; e$t varius
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1151 motus formic&aelig; pendet ex latione plani, ide&ograve; velocitates
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1152 eiu$dem (nam in plano immobili ponimus &aelig;quabiliter fer-
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1153 ri) durant ij$dem temporibus, quibus velocitates pr&aelig;cipu&aelig;
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1154 gene$is ACBH. Sit denique LMSR imago velocitatum
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1155 iuxta gene$im ACBH, cuius temporis imago CD &amp; B; pa-
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1156 tet $i $it MP ad PS $icut imago temporis CDEG ad ima-
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1157 ginem &amp; BGE &amp;, fore LM ad PQ vt AC ad OG, &amp; con-
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1158 cepta etiam figura MNOTS inter parallelas LMN, RST
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1159 ita vt $it $emper MN ad PO $icut FC ad GI, nec non LM
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1160 ad MN vt AC ad FC. ($unt enim initio motuum in C, aut
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1161 in$tanti M, velocitates gene$ium AC, CF, $cilicet LM, MN;
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1162 &amp; in G, hoc e$t in$tanti P $unt velocitates OC, GI; nimi-
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1163 rum QP, PO) vocetur proinde gene$is FCBK $puria, ac
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1164 ad$tricta imaginitemporis &amp; DCB &amp;, cuius imago veloci-
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1165 tatum MNTS pariter $puria, homogenea tamen ip$i legiti-
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1166 <*>&aelig; LMSR.
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1167 <pb n=33>
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1168 <C>PROP. XVI. THEOR. XVI.</C>
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1169 <p>SI $int duo motus iuxta gene$es legitimam, &amp; $puriam,
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1170 erunt mobilium exacta $patia, vt imagines inter$e
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1171 homogene&aelig; velocitatum, legitima ad $puriam.
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1172 <p>E$to gene$is legitima ACBH, cuius imago temporis
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1173 <MARG><I>Tab.</I> 3. <I>Fig.</I> 4.</MARG>
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1174 &amp; DCA &amp;, &amp; imago velocitatum MLRS. Sit etiam gene-
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1175 $is altera illi homogenea, $ed $puria, &amp; ad$tricta imagini
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1176 temporis &amp; DCB &amp;, cuius imago velocitatum $puria, prio-
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1177 rique legitim&aelig; homogenea NMST. Dico, $patia iuxta has
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1178 imagines tran$acta e$$e vt ip$&aelig; imagines legitima LMSR
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1179 ad $puriam NMST. Cum temporis momenta M, P in-
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1180 telligantur ex minimis temporibus, qu&aelig; proponi po$$unt,
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1181 inter$e &aelig;qualibus, &amp; quibus &aelig;quabiliter perdurant ve-
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1182 locitates, quas mobile $ortitur in aduentu $uo in punctis
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1183 C, G, erit vt velocitas FC ad velocitatem GI $ic inter$e
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1184 <MARG><I>Pr.</I> 3. <I>huius.</I></MARG>
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1185 $patia, qu&aelig; i$tis velocitatibus, temporibu$que illis &aelig;qua-
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1186 libus percurrerentur, in qua ratione e$t etiam NM ad OP.
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1187 Deinde momento M peragerentur $patia proportionalia
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1188 velocitatibus FC, AC, $eu rectis NM, ML, momento
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1189 autem P $patia proportionalia velocitatibus GI, GD,
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1190 in qua ratione e$t etiam OP ad PQ, &amp; $ic deinceps
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1191 procedendo per $ingula temporis MR momenta, adeo
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1192 vt, cum $patium velocitate FC exactum ad id veloci-
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1193 tate CA, $it vt NM ad ML, $patium velocitate IG ad id
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1194 exactum velocitate GD $it vt OP ad PQ, &amp; $int pr&aelig;terea
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1195 prim&aelig; inter$e, hoc e$t $patia velocitatibus FC, GI tran-
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1196 $acta, proportionalia tertijs, $patijs videlicet tran$actis
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1197 velocitatibus ML, PQ ergo vt omnes prim&aelig; ad omnes
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1198 tertias quantitates, hoc e$t omnia $patia tran$acta iuxta
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1199 gene$im FCBK ad omnia $patia iuxta gene$im ACB, ita
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1200 erit $umma $ecundarum ad &verbar;omnes quartas, $cilicet i$ta
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1201 erit imago NMST ad imaginem LMSR. Quod &amp; c.
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1202 <pb n=34>
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1203 <C>LIBER ALTER</C>
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1204 <C>DE</C>
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1205 <C>Motu Compo$ito.</C>
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1206 <p>MOtum appellamus compo$itum, vbi dum fer-
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1207 tur mobile, con$ideratur habere plures i&ntail;
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1208 diuer$as partes, vel eti&atilde; in eandem partem
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1209 conatus, ex quibus oriatur tertia vis di$tin-
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1210 cta ab illis. Hunc librum, cum expleueri-
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1211 mus, non pauca vn&agrave; cum priori, dicta erunt de motu, erit-
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1212 que ea methodus, qua $imul geometrica qu&aelig;dam, difficil-
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1213 lima $citu $atis breuiter o$tendemus. Nam vibrationes
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1214 pendulorum exigi temporibus; qu&aelig; $int in $ubduplicat&atail;
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1215 ratione longitudinum eorundem, plan&egrave; tandem con$tabit
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1216 ali&agrave;s nobis di$$entientibus: aperiemus etiam, qua arte in-
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1217 telligi queant anguli rectilinei curuilineis &aelig;quales; nec non
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1218 exponemus parabolas quibu$dam $piralibus &aelig;quales, vt
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1219 e$t vulgata $pirali Archimede&aelig;, c&ugrave;m videlicet ba$is para-
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1220 bol&aelig; radio circuli $piralem continentis, &amp; dimidium huius
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1221 circumferenti&aelig; circuli altitudini eiu$dem parabol&aelig;, &aelig;qua-
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1222 les $int.
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1223 <C>PROP. I. THEOR. I.</C>
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1224 <MARG><I>Tab.</I> 4. <I>Fig.</I> 1.</MARG>
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1225 <p>SI in eadem recta linea currantur $patia temporibus
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1226 &aelig;qualibus, &amp; $int motus $implices, ac ad ea$dem par-
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1227 tes tendentes, eadem illa $patia $imul motu compo$ito, ab
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1228 eodemque mobili duabus illisgene$ibus affecto, vnicoque
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1229 ex dictis temporibus &aelig;qualibus, excurrentur.
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1230 <pb n=35>
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1231 <p>Curratur LI iuxta imaginem velocitatum HAEF, et IO
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1232 iuxta aliam dict&aelig; homogeneam BAED. Dico LO $um-
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1233 mam dictorum $patiorum LI, IO exactum iri vnico tem-
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1234 pore AE, $i nempe mobile feratur $ec&utilde;dum vtranque ima-
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1235 ginem.
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diff changeset
1236 <p>Per quodlibet punctum, $eu temporis momentum M
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1237 agatur recta GMC parallela HB, vel FD. Habebit mobi-
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1238 le momento A, d&utilde; $cilicet mouetur motu compo$ito duas
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1239 $imul velocitates AH, AB, ide$t vnicam HB. Similiter mo-
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1240 mento M habebit GC, &amp; momento E ip$am FD. Itaque
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1241 <MARG><I>Def.</I> 3. <I>prima.</I></MARG>
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1242 erit HBDF imago velocitatum compo$iti motus, qui fiet
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1243 tempore AE iuxta imaginem, qu&aelig; aggregatum e$t dictar&utilde;
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1244 HAEF, ABDE. E$t ver&ograve; LI ad IO vt imago HAEF ad
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1245 imaginem ABDE; ergo conuertendo, componendoqu&etail;
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1246 erit vt LI ad LO, $ic imago HAEF ad imaginem HBDF;
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1247 propterea quemadmodum $patium LI currebatur iuxt&atail;
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1248 imaginem HAEF, $ic LO percurretur imagine HBDF $olo,
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1249 eodemque tempore AE. Quod &amp;c.
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1250 <C><I>Corollarium.</I></C>
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1251 <p><I>Hinc patet graue perpendiculariter, violenterque deiectum
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1252 minim&egrave; ad terram venturum aggregato virium, quarum vna
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1253 e$t ab impellente impre&szlig;a, altera ver&ograve; &agrave; grauitate depend&etilde;s.
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1254 Nam ex impartita vt celerior fit ca$us, quam vt graue in de-
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1255 cur$u $uo po$$it ex acceler atione naturali eum gradum acqui-
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1256 rere, quem cert&egrave; $ponte $ua tant&ugrave;m de$cendens in fine eiu$dem
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1257 altitudinis adeptum e$$et. Hoc ita verum e$t, vt aliquando
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1258 minimum inter$it, inter impetum ab ambabus cau$is proue-
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1259 nientem, &amp; eum, qui a $ola oritur grauitate, quamobrem pa-
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1260 rum is proficeret, qui conaretur maiorem impetum componere
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1261 in ca$u grauis, illi nempe adiecta vi, mobile idem in decur$u
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1262 impellente, vltra nat<*>am grauitatem, quod tamen fieri haud
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1263 dubi&egrave; po$$et, $i ca$us obliquus e&szlig;et.</I>
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1264 <pb n=36>
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1265 <p><I>Illud quoque hac occa$ione aperiendum e$t, graue naturali-
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1266 ter de$cendens e&ograve; concitati&ugrave;s ferri, quoad potentia re$i$tentis
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1267 aeris (validior namque i$ta fit, vbi mobilis ca$us e$t celerior)
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1268 vi granitatis mobili inh&aelig;renti exaquatur, tunc enim cau$&atail;
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1269 vlterioris accelerationis adempta e$t, con$umiturque in lucta-
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1270 tione aeris contranitentis: quare tunc grane progrederetur
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1271 aquabili motu, id qu&ograve;d citi&ugrave;s euenire deberet $i grane intr&atail;
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1272 aquam de$cendat.</I>
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1273 <C>PROP. II. THEOR. II.</C>
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1274 <p>SI in eadem recta duos motus $ibi contrarios, fimplices,
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1275 ac eodem tempore peractos intelligamus, mobile di-
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1276 ferentiam illorum $patiorum, $i vtroque motu e$$et affe-
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1277 ctum, percurreret.
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1278 <MARG><I>Tab.</I> 4. <I>Fig.</I> 2.</MARG>
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1279 <p>Curratur &agrave; puncto L $patium LO imagine velocitatum
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1280 ABFG, &amp; codem tempore curratur etiam recta OM ex
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1281 puncto altero O, $cilicet contrario motu, &amp; iuxta imagin&etilde;
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1282 AHIG pr&aelig;dict&ecedil; homogeneam. Dico mobile, c&otilde;po$ito ex
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1283 vtri$que motu, &amp; tempore ip$o AG cur$urum differentiam
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1284 LM dictorum $patiorum LO, OM.
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1285 <p>Prim&ugrave;m intra parallelas AB, GF non $e $ecent line&aelig;
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1286 <MARG><I>Tab.</I> 4. <I>Fig.</I> 2.</MARG>
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1287 BF, HI, &amp; ducatur qu&aelig;libet DC &aelig;quidi$tans AB, vel GF,
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1288 qu&aelig; fecet HI in E. Manife$tum e$t, mobile, compo$ito
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1289 motu feratur habere duplicem velocitatem, vnam AB al-
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1290 teram illi oppo$itam AH, ob idque moueri ver$us O $ol&atail;
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1291 velocitate HB differentia dictarum inter$e pugnantium
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1292 velocitatum: pariter momento D feretur mobile veloci-
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1293 tate EC differentia duarum DE, DC, &amp; in$tanti G habebit
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1294 <MARG><I>Def.</I> 3 <I>prima.</I></MARG>
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1295 differentialem IF; ex quo $equitur figuram BHEIFCB, dif-
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1296 ferentiam imaginum ABFG, HAGI, aptatam tempori AC
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1297 imaginem e$$e velocitatum compo$iti motus. Hoc po-
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1298 <MARG><I>Pr.</I> 2. <I>prim&atail;
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1299 huius.</I></MARG>
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1300 $ito habebit LM ad LO eandem rationem, ac BHIF ad
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1301 ABFG; Propterea LM, qu&aelig; e$t differentia $patiorum LO,
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1302 <pb n=37>
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1303 MO curretur iuxta imaginem BHIF, nempe compo$ito
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1304 motu, &amp; tempore AG.
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1305 <p>2. Se nunc $ecent line&aelig; BF, HI in C. Ducatur CD pa-
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1306 <MARG><I>Tab.</I> 4. <I>fig.</I> 3.</MARG>
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1307 rallela alteri &aelig;quidi$tantium AB, GF. Con$tat ex prima
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1308 parte, qu&ograve;d mobile compo$ito motu, &amp; iuxta imaginem
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1309 HBC feretur ver$us O tempore AD; $it ergo $patium, quod
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1310 curreretur illa imagine, PR, &amp; ob id LO ad PR eande&mtail;
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1311 <MARG><I>Pr.</I> 2. <I>prima</I></MARG>
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1312 habebit rationem quam imago ABFG ad imagine&mtail;
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1313 HBC.
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1314 <p>Similiter dum mobile mouetur tempore DG iuxta ima-
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1315 gines DCIG, DCFG, feretur ver&egrave; $ecund&ugrave;m imagine&mtail;
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1316 <MARG><I>Ex prim&atail;
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1317 parte.</I></MARG>
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1318 FCI ver$us L, quamobrem $i $patium, quod exigeretur
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1319 hac imagine $it RQ, habebit i$tud ad LO eandem rationem,
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1320 <MARG><I>Pr.</I> 2. <I>prima.</I></MARG>
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1321 quam imago CFI ad imaginem ABFG, &amp; ideo ex &aelig;quali
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1322 QR ad PR $e habebit vt imago CFI ad imaginem HBC; $i
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1323 igitur ponatur ABFG maior imagine AHIG, dempt&agrave; co-
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1324 muniter AHCFG relinquetur HBC maior imagine CEI, &amp;
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1325 ideo etiam PR maior QR: curritur ver&ograve; PR vers&ugrave;s R tem-
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1326 pore AD, &amp; RQ vers&ugrave;s P tempore DG, ergo toto tempo-
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1327 re AG curretur PQ differentia $patiorum PR, RQ. Cum
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1328 ver&ograve; HBC ad CFI, $it vt PR ad RQ, erit diuidendo vt ex-
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1329 ce$$us imaginis HBC $upra imaginem FCI ad imagine&mtail;
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1330 i$tam, ita PQ ad QR, &amp; o$ten$um e$t QR ad LO, $icut ima-
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1331 go FCI ad imaginem ABFG, ergo ex &aelig;quali exce$$us ima-
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1332 ginis HBC $upra imaginem AHIG habebit eandem ratio-
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1333 nem ad imaginem AHIG, ac PQ ad LO, at e$t in illa ead&etilde;
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1334 ratione etiam LM ad LO (e$t enim LO ad MO vt imago
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1335 ABFG ad imaginem AHIG) ergo PQ erit &aelig;qualis LM,
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1336 atque adeo mobile dum currit vtroque motu, hoc e$t iux-
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1337 ta $imul duas imagines propo$itas contrariorum motuum,
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1338 peraget $patium LM vers&ugrave;s O $ecund&ugrave;m imaginem, qu&aelig;
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1339 differentia e$t propo$itarum ABFG, AHIG, tempore AG.
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1340 Quod &amp;c.
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1341 <pb n=38>
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1342 <C><I>Corollarium.</I></C>
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1343 <p><I>Deduc&igrave;tur, mobile nullum $patium emen$urum, vbi ima-
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1344 gines $implicium motuum fuerint aquales.</I>
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1345 <C>PROP. III. THEOR. III.</C>
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1346 <MARG><I>Tab.</I> 4. <I>Fig.</I> 4.</MARG>
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1347 <p>REperire eam velocitatem, eamque directionem, qu&aelig;
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1348 orirentur, $i mobile pluribus eodem momento velo-
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1349 citatibus, $eu conatibus affectum e$$et. Opportet autem
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1350 non $olum has velocitates, ver&ugrave;m etiam earum directio-
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1351 nes manife$tas e$$e.
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1352 <p>Habeat mobile A, eodem momento conatum AB, quo
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1353 tendat in R; AC; quo in C; &amp; AD, quo in D. Qu&aelig;ritur ve-
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1354 locitas, &amp; directio, quas mobile habiturum e$$et in multi-
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1355 plici illa affectione (Nam actu vnam velocitatem, vnam-
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1356 que tant&ugrave;m directionem $ortiri debet) Ex duabus qui-
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1357 bu$que AD, AC intelligatur perfici parallelogrammum
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1358 ACED, &amp; ducta diametro AE fiat itidem aliud parallelo-
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1359 grammum ABFE, cuius agatur diameter AF. Dico AF
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1360 e$$e qu&aelig;$itam velocitatem, ac directionem, quibus mobile
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1361 ex illis pluribus conatibus motum $uum in$titueret.
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1362 <p>Si mobili A currendum e$$et &aelig;quabili motu $patium
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1363 AE, pertran$iret eodem tempore tam rectam AD, qu&agrave;m
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1364 <MARG><I>C<*>l. pr. de mo-
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1365 <*> aquab.</I></MARG>
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1366 ip$am AC; nam cum fertur ab A in E ver&egrave; de$cendit ab A
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1367 in C, &amp; ab A in D motu pariter &aelig;quabili; ergo AD ad
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1368 AC, erit vt velocitas, qua curritur per AD ad velocitatem,
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1369 qua curritur per AC. Itaque $i mobile dum e$t in A in-
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1370 telligatur affectum velocitatibus AD, AC habentibus di-
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1371 rectiones ip$as rectas AD, AC, perinde e$$et, ac $i $ola fo-
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1372 ret mobili velocitas vn&acirc; cum directione AE. Eadem ra-
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1373 tione AF velocitas habens directionem AF, &aelig;quipollebit
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1374 duabus velocitatibus AB, AE iuxta directiones rectas ea$-
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1375 <pb n=39>
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1376 dem ABAE; hoc &aelig;quiualebit tribus AB, AC, AD. Mo-
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1377 bile igitur ex affectione trium illorum conatuum, vt $up-
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1378 po$itum fuit, nitetur $ecund&ugrave;m AF velocitate ip$a AF
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1379 Quod &amp;c.
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1380 <C>DEF. I.</C>
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1381 <p>ACcelerationem alicuius motus, tunc intelligimus, c&utilde;
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1382 velocitates, qu&aelig; $ubinde mobili adueniunt, non de-
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1383 lentur, $ed pror$us integr&aelig;, atque indelebiles mobili in ip$o
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1384 motu per$euerant. Ex quo $equitur motum $implicem di-
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1385 ci, cum pr&aelig;terit&aelig; velocitates protinus euane$cunt, ill&aelig;-
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1386 que tantum con$iderantur, qu&aelig; mobili $ubinde oriun-
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1387 tur.
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1388 <C>PROP. IV. PROB. II.</C>
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1389 <p>IMaginem accelerationis cuiu$cunque $implicis motus
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1390 exhibere.
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1391 <p>Imago velocitatum $implicis motus e$to rectangulum
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1392 <MARG><I>Tab.</I> 4. <I>fig.</I> <*>.</MARG>
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1393 AFDC: $ic motus e$t &aelig;quabilis, vt acceleretur debent in-
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1394 <MARG><I>Cor. def.</I> 3. <I>pri-
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1395 mi.</I></MARG>
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1396 $tanti C vigere omnes velocitates in imagine AFDC c&otilde;-
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1397 prehen$&aelig;, &amp; item ducta quacunque BE parallela AF, vel
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1398 <MARG><I>Def.</I> 1. <I>huius.</I></MARG>
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1399 CD, erit mobile momento B affectum omnibus antece-
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1400 dentibus velocitatibus, comprehen$is nempe ab imaginis
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1401 portione AFEB; quare $i ponamus HLG imaginem e$$&etail;
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1402 accelerationis, itaut nempe tempus GL &aelig;quale $it tempo-
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1403 ri AC; item KL &aelig;quale tempori AB, erit vt figura CAFD
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1404 ad figuram BAFE, $ic velocitas, qua mobile fertur mom&etilde;-
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1405 to G ad velocitatem, quam habet in$tanti K; &amp; ideo quia
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1406 ponitur imago $implicis motus rectangulum AFDC, erit
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1407 rectangulum CF ad BF, hoc e$t recta CA ad AB imm&ograve;
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1408 LG ad LK, vt GH ad KI; quamobrem GLH imago velo-
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1409 <MARG><I>Def.</I> 3 <I>primi.</I></MARG>
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1410 citatum huiu$modi motus, erit triangulum. Quod $i ima-
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1411 <pb n=40>
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1412 go $implicis motus fui$$et triangulum, imago velocitatum
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1413 accelerationis foret trilineum $ecundum, &amp; ita pro-
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1414 portionaliter de infinitis numero accelerationibus.
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1415 <C><I>Corollarium.</I></C>
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1416 <p><I>Hinc obiter habemus, quo pacto imago velocitatum corpo-
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1417 rum naturaliter de$cendentium triangulum $it. Nam quo-
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1418 libet momento $ui ca$us habet graue idem in$e principiu&mtail;
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1419 motus, $eu grauitas, ex qua concipitur imago $implicis motus
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1420 $i nempe priores gradus velocitatis $ubinde deperirent, at
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1421 quia in eius de$cen$u pror$us per$euerant (id enim $upponi-
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1422 tur ab$trabendo ab aere) inde motus concitatur, &amp; fit vti di-
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1423 ximus imago accelerationis triangulum.</I>
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1424 <C>AXIOMA</C>
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1425 <p>QV&aelig;libet linea, vt fluxus puncti concipi po-
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1426 te$t.
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1427 <C>AX. II.</C>
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1428 <p>VT propo$ita linea ex fluxu puncti exar&egrave;tur, du&ograve; tan-
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1429 t&ugrave;m nece$$aria $unt, $cilicet motus, &amp; puncti di-
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1430 rectio.
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1431 <C>PROP. V. THEOR. III.</C>
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1432 <p>REcta, qu&aelig; pri&ugrave;s de$eripta e$t, pote$t alijs &agrave; primis
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1433 velocitatibus, rur$us exarar<*>.
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1434 <p>Nam punctum pote$t fluere $ecundum quamcunque
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1435 rectam, quocunque motu, ergo illam pote$t etiam quibu$-
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1436 cunque velocitatibus affectum rur$us exarare.
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1437 <pb n=41>
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1438 <C>PROP. VI. THEOR. IV.</C>
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1439 <p>VT eadem recta ex fluxu puncti renouetur, opportet in
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1440 quocunque illius puncto $eruari pri$tinas directio-
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1441 nes,
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1442 <p>Cum, vti diximus, ad de$criptionem line&aelig; duo tant&ugrave;m
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1443 <MARG><I>Ax.</I> 2. <I>buius.
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1444 pr.</I> 5. <I>huius.</I></MARG>
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1445 exigantur, nempe motus, &amp; puncti directio; motus ver&ograve; po-
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1446 te$t e$$e quilibet, $equitur ergo directionem, alteram de
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1447 duobus, $eruari debere.
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1448 <C>DEF. II.</C>
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1449 <p>LIneam dicimus curuam, in qua $umptis duobus ad-
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1450 libitum punctis, recta, qu&aelig; ip$a puncta coniunge-
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1451 ret, nullam cum propo$ita linea partem $it habitura com-
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1452 munem.
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1453 <C>PROP. VII. THEOR. V.</C>
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1454 <p>DIrectiones puncti de$cribentis lineam, iuxta rectas
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1455 lineas concipi debent.
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diff changeset
1456 <p>Dum punctum fluere intelligimus, ine$t in eo $ingulis
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diff changeset
1457 momentis certus, ac pr&aelig;fixus gradus velocitatis, quo tan-
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diff changeset
1458 t&ugrave;m attento, rect&agrave;, &aelig;quabiliq; motu in certam partem con-
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diff changeset
1459 tenderet; at huiu$modi iter, aliud non e$t, qu&agrave;m directio
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diff changeset
1460 puncti, qua eius temporis momento profici$citur; ergo iux-
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diff changeset
1461 ta rectas lineas, directiones omnes con$iderari opportet.
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diff changeset
1462 <C>PROP. VIII. THEOR. VI.</C>
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diff changeset
1463 <p>TAngens, &amp; directio motus in quouis curu&aelig; puncto
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diff changeset
1464 e$t vna, atq; eadem recta.
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diff changeset
1465 <p>Nam in de$criptione cuiu$cunq; rect&aelig; procedit pun-
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diff changeset
1466 <MARG><I>Pr.</I> 7. <I>huius.</I></MARG>
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diff changeset
1467 <pb n=42>
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1468 ctum iuxta tendentias rectas, obliquatur tamen ob $ub$e-
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diff changeset
1469 quentes, ali&ograve; tendentes ni$us, &amp; ob id di$trahitur punctum
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diff changeset
1470 ip$um &agrave; priori tendentia, idem accidit ex alia parte $i re-
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diff changeset
1471 flaxiffet idem punctum, nempe hinc inde vnicam rectam
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diff changeset
1472 eandemque, continuantibus oppo$itis ad idem punctum
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diff changeset
1473 directionibus, ergo directio, &amp; tangens vna, &amp; eadem e$t
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1474 recta.
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diff changeset
1475 <C><I>Corollarium.</I></C>
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1476 <p><I>Hinc $equitur, vnicam lineam dicendam e$$e, cum &agrave; quo-
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diff changeset
1477 cunque illius puncto vnica tant&ugrave;m ex vtraque parte egre-
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diff changeset
1478 ditur tangens.</I>
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diff changeset
1479 <C>DEF. III.</C>
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1480 <p>QV&ograve;d $i ex aliquo puncto du&aelig; tangentes hinc inde
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1481 egredientes angulum efficiant; tunc propo$itam li-
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diff changeset
1482 neam inflexam dicemus, &amp; punctum, in quo $unt
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diff changeset
1483 contactus, inflexionis appellabitur.
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diff changeset
1484 <C><I>Corollarium I.</I></C>
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1485 <p><I>Ab hi$ce deffinitionibus, &amp; priori coroll. manat artificium
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1486 componendi duas curuas, vel curuam &amp; rectam, adeout vni-
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1487 cam lineam efforment, nullumque angulum; nempe cum $ic
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diff changeset
1488 inuicem tuxgamus, vt taugentes ad punctum connexus, vnam
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1489 tant&ugrave;m rectam efficiant.</I>
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diff changeset
1490 <C><I>Corollarium II.</I></C>
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1491 <p><I>Sed &amp; illud patet, quibus angulis inflectantur line&aelig; inui-
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1492 cem compo$it<*>, $i ad punctum inflexionis angulum tangen-
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1493 tium ob$eruauerimus, $unt enim inter$e &aelig;quales, lic&egrave;t diuer-
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1494 $a $peciei, <*> vnus $it curuilineus, &amp; rectilineus alter.</I>
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1495 <pb n=43>
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diff changeset
1496 <C>PROP. IX. THEOR. VII.</C>
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1497 <p>TAngens, $eu directio motus in quocunque curu&aelig;
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1498 puncto e$t illa recta, qu&aelig; vtrinque $tatim cadens
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1499 extra curu&aelig; conuexum ad eandem, qu&agrave;m fieri pote$t ex
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1500 vtraque parte accedit.
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1501 <p>Nam alia qu&aelig;que recta tran$iens per punctum conta-
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1502 ctus ad $ectionem magis accedere nequit, quin ip$am illinc
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1503 $ecet, ob id extra conuexum eius non cadet, ab altera ve-
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1504 r&ograve; parte magis &agrave; propo$ita curua $eparabitur, quamobrem
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1505 nulla alia recta, qu&agrave;m tangens poterit $imul extra curuam
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1506 e$$e, &amp; qu&agrave;m fieri pote$t ad ip$am accedere.
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1507 <C>DEF. IV.</C>
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1508 <p>LIne&aelig; AC, AD occurrant $ibi in A, quod punctum in-
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1509 <MARG><I>Tab.</I> 4. <I>fig.</I> 6.</MARG>
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1510 telligatur transferri ab A in C vn&agrave; cum linea AD
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1511 $emper $ibi parallela, quo tempore punctum A currat ip-
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1512 $am latam lineam ex A in D. Manife$tum e$t idip$um
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1513 punctum A de$cripturum e$$e motu compo$ito lineam
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1514 quandam AB diagonalem $uperficiei parallelogramm&aelig;
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1515 ABCD. Vocamus ergo diagonalem illam $emitam com-
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1516 po$iti motus, &amp; AC, AD latera illius.
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1517 <C><I>Corollarium I.</I></C>
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1518 <p><I>Manife$tum e$t mobile dum currit AB tran$ire etiam AC,
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1519 AD, lic&egrave;t curu&aelig; $int, nam ver&egrave; transfertur illo tempore, tam
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1520 ad lineam CB quam ad DB.</I>
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1521 <C><I>Corollarium II.</I></C>
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1522 <p><I>Pr&aelig;terea $i ducerentur, aut$int AC, CB, DA, DB, AB</I>
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1523 <pb n=44>
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1524 <I>rect&aelig; line&aelig;, efficeretur ex ijs parallelogrammum ACBD, cu-
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1525 ius diameter AB; quamobrem ex datis punctis C, A, D repe-
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1526 riretur $tatim punctum B, $cilicet extremum $emit&aelig; compo-
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1527 $iti motus, cuius latera ip$&aelig; curu&aelig;, aut rect&aelig; AC, AD</I> &mdash;.
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1528 <C>PROP. X. PROB. III.</C>
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1529 <p>EX datis quotcunq; lateribus compo$iti motus, huius
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1530 <MARG><I>Tab.</I> 4. <I>Fig.</I> 7.</MARG>
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1531 $emit&aelig; terminum exhibere.
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1532 <p>Si latera compo$iti motus e$$ent duo tant&ugrave;m AB, AC.
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1533 Facto parallelogrammo vt dictum e$t, inueniretur pun-
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1534 ctum E extremum motus: &amp; qu&aelig;cunq; $it $emita, $eu mo-
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1535 tus, pote$t idem E $upponi tanquam extremum alterius la-
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1536 teris, adeoque, $i motus con$tet ex tribus lateribus AC,
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1537 AB, AD, perinde $it ac $i foret duorum laterum AE, AD;
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1538 nam AC, AD valent $imul ac $olum AE; cum ita $it, facto
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1539 etiam parallelogrammo EADF ex datis punctis E, A, D,
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1540 habebitur F extremum $emit&aelig;, cuius $unt tria latera CA,
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1541 AD, AB &mdash;
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1542 <C><I>Corollarium.</I></C>
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1543 <p><I>Deducitur artificium de$cribend&aelig; $emit&aelig; AE, vel AF, $i
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1544 nempe a$$umptis partibus AG, AH, AI in dictis lateribus,
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1545 qu&aelig; quidem $ciantur percurri temporibus &aelig;qualibus, $i per
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1546 ip$as $ingulas mobile punctum ferretur eo modo, quo in com-
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1547 po$ito motu nititur per ea$dem directiones; reperietur in-
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1548 quam punctum K in $emita AE, atque L in $emita AF: qua-
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1549 re hoc modo $umptis alijs, atque alijs partibus in ip$is lateri-
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1550 bus, reperientur alia, atque alia puncta ad ip$am $emita&mtail;
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1551 pertinentia, quorum tandem beneficio, facile erit qua$itam
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1552 ferm&egrave; $emitam exarare.</I>
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1553 <pb n=45>
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1554 <C>PROP. XI. PROB. IV.</C>
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1555 <p>EX datis imaginibus velocitatum, iuxta quas $implici
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1556 <MARG><I>Tab.</I> 4. <I>fig.</I> 8.</MARG>
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diff changeset
1557 motu currantur latera compo$iti motus; datis item
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1558 tangentibus ad qu&aelig;cunque puncta ip$orum laterum, repe-
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1559 rire $emitam compo$iti motus, nec non directiones, veloci-
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1560 tate$q; puncti de$cribentis ip$am $emitam.
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diff changeset
1561 <p>Opportet tamen latera ip$a, itemq; imagines pr&aelig;dictas,
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1562 in imperatas $ecari po$$e rationes, quamquam nos non la-
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diff changeset
1563 teat, in lateribus curuis hoc effici non po$$e, pr&aelig;terqua&mtail;
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1564 aliquatenus in periph&aelig;rijs circulorum.
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diff changeset
1565 <p>Sint AB, AF latera compo$iti motus, qu&aelig; quidem $eor-
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diff changeset
1566 $im currantur eodem tempore QM, $cilicet AB iuxta ima-
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diff changeset
1567 ginem MNPQ, et AF iuxta imaginem alteram ei homoge-
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diff changeset
1568 neam TMQR. Ponatur AB circuli arcus, quem tangat re-
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diff changeset
1569 cta BC &aelig;qualis QB, at AF lineam', qu&aelig; parabola $it, con-
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diff changeset
1570 tingat recta FG &aelig;qualis RQ. Reperiemus illic&ograve; punctum
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1571 <MARG><I>Pr,</I> 10. <I>huius.</I></MARG>
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diff changeset
1572 H extremum $emit&aelig; compo$iti motus; $unt enim data pun-
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diff changeset
1573 cta A, F, B. Cum igitur mobile venerit in H. Dico, eo
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diff changeset
1574 temporis momento velocitatem, ac directionem HL, qu&aelig;
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1575 recta diameter e$t parallelogrammi, cuius duo latera $unt
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1576 dict&aelig; line&aelig; HI, HK; Iam vti diximus punctum H e$t ex-
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1577 tremum compo$iti motus, quare eo momento, quo pun-
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diff changeset
1578 ctum mobile e$t in H, habet inibi ea$dem illas velocitates,
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diff changeset
1579 quas haberet in B, et F, dum $eor$im illa latera excurri$$et;
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diff changeset
1580 $cilicet con$ideratur ip$um mobile habens $imul velocita-
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diff changeset
1581 tem HI &aelig;qualem, ac &aelig;quedirectam, $eu &aelig;quidi$tantem
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1582 ip$i CB, cui e$t &aelig;qualis alia QP; &amp; velocitatem HK &aelig;qua-
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1583 lem, $imiliterque directam, ip$i GF &aelig;quali RQ. Cum ita
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1584 <MARG><I>Ex pr.</I> 3. <I>hu.</I></MARG>
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1585 $it erit HL velocitas, &amp; directio qu&aelig;$ita momento Q. Eo-
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1586 dem modo, $i $it, vel fiat vt imago PNMQ ad ONMV
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1587 (ducta $cilicet applicata SVO) ita BA ad AX, et ONMV
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diff changeset
1588 ad imaginem VMTS, vt XA ad AI, percurrentur AX, AI
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1589 <MARG><I>Pr.</I> 2. <I>primi
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1590 huius.</I></MARG>
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1591 <pb n=46>
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1592 eodem tempore MV, eritque ob id in X velocitas, &amp; dire-
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1593 ctio, tangens ip$a ZX &aelig;qualis VO, &amp; in I velocitas, &amp; di-
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1594 rectio, tangens 2 I &aelig;qualis VS; Itaque datis punctis X, I, A
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1595 <MARG><I>Cor.</I> 2. <I>def.</I> 3.
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1596 <I>huius.</I></MARG>
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1597 dabitur etiam Y extremum $emit&aelig; compo$iti motus, cuius
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1598 latera AX, AI, &amp; ideo mobile dum e$t in Y momento V
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1599 affectum erit duplici velocitate, hoc e$t Y 4 &aelig;quali ve-
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1600 locitati ZX, $eu VO, ac &aelig;quidi$tante eidem ZX, et veloci-
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1601 tate altera Y 3 &aelig;quali, &amp; &aelig;qu&egrave;directa ip$i 2 I: quare ex
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1602 datis punctis 4, Y, 3 inuenietur punctum S quartus angu-
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1603 lus parallelogrammi habentis diametrum YI, qu&aelig; quidem
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1604 <MARG><I>Pr.</I> 3. <I>huius</I></MARG>
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1605 erit directio, &amp; velocitas mobilis currentis compo$ito mo-
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1606 tu in$tanti V. Cumque alia quotcunque puncta eadem
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1607 methodo reperire queamus, per qu&aelig; duci po$$it linea fer&egrave;
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1608 qu&aelig;$itam $emitam repr&aelig;$entans, atq; emulans, patet idcir-
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1609 co, quod propo$uimus.
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1610 <C><I>Corollarium.</I></C>
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1611 <p><I>Cum ver&ograve; directiones $int idem, ac tangentes, liquet HL</I>
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1612 <MARG><I>Pr.</I> 8 <I>huius.</I></MARG>
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1613 <I>VS tangentes e$$e compo$iti motus.</I>
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1614 <C>PROP. XII. THEOR. VIII.</C>
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1615 <p>CVm imagines velocitatum, iuxta quas curruntur du&ecedil;
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1616 <MARG><I>Tab.</I> 5. <I>Fig.</I> 1.</MARG>
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1617 rect&aelig;, qu&aelig; $int latera compo$iti motus, $unt paral.
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1618 logrammum, &amp; triangulum; tunc $emita compo$iti motus
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1619 erit communis parabola.
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1620 <p>Tempore HM curratur latus AC iuxta imaginem velo-
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1621 citatum HILM rectangulum, &amp; latus AB iuxta imaginem
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1622 <MARG><I>pr.</I> 2. <I>primum
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1623 huius.</I></MARG>
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1624 triangulum HMN; erit CA ad AB, vt imago parallelogr&atilde;-
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1625 mum HILM ad aliam imaginem triangulum NHM. Fiat
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1626 <MARG><I>Pr.</I> 3. <I>huius.</I></MARG>
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1627 parellogr&atilde;mum ACDB erit in D extremum $emit&aelig; com-
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1628 po$iti motus, qu&aelig; $i ponatur AFC; Dico e$$e parabolam.
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1629 Sumatur in ip$a linea quoduis punctum F, ab ip$o dedu-
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1630 <pb n=47>
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1631 cta FE parallela AB, vti etiam FG parallela AC, erunt
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1632 <MARG><I>Ex eadem.</I></MARG>
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1633 AE, AG latera compo$iti motus, cuius $emita AF: Con-
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1634 cipiatur mod&ograve; P momentum, quo mobile ade$t in F, &amp;
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1635 ducta OPK parallela alteri HI, vel NL, erit imago MHIL ad
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1636 <MARG><I>Pr.</I> 2. <I>huius.</I></MARG>
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1637 imagin&etilde; PHIK, hoc e$t MH ad HP, vt CA ad AE, $eu vt BD
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1638 ad GF. Pariter erit imago NHM ad imagin&etilde; OHP, hoc e$t
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diff changeset
1639 quadratum ex MH ad quadrat&utilde; ex PH; imm&ograve; id ex BO ad
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diff changeset
1640 illud ex GF, vt BA ad AG; quamobrem punctum F cadet
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1641 in curuam parabolicam communem, cuius diameter AB,
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1642 &amp; ba$is, $eu ordinatim applicata BD, $cilicet AFD erit ip$a
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1643 curua parabolica. Quod &amp;c.
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1644 <C><I>Scholium.</I></C>
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1645 <p><I>Quoniam graue, quod iaculatur extr&aelig; perpendiculum, li-
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1646 berum ab omni obice, ni$i turbaretur eius motus &agrave; propri&atail;
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diff changeset
1647 grauitate pergerct moueri &aelig;quabiliter iuxta directionem, ve-
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1648 locitatemque ei traditam; habet ver&ograve; coniunctam grauita-
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1649 tem, qua, ni$i ab impre$$o impetu flecteretur motus, de$cen-
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1650 deret iuxta perpendiculum motu naturaliter concitato, cuius
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1651 imago velocitatum, triangulum e$t; Hinc propterea gran&etail;
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1652 vltra perpendiculum proiectum de$cribit in cur$u $uo, motu
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diff changeset
1653 $cilicet compo$ite, parabolam vulgatam. Ver&ugrave;m enim ver&ograve;
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1654 de$criptionem i$t am nece$$e aliquo pacto e$t ex duabus cau$is
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1655 vitiari, hoc est ab aeris re$i$tentia, &amp; perpendiculis non in-
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1656 ter$e parallelis, quippe in idem, vnumq; punctum, vniuer$i
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1657 centrum, conuergentibus.</I>
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1658 <C>PROP. XIII. THEOR. IX.</C>
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1659 <p>SI ab a$$umpto hyperbol&aelig; puncto, recta axi primo pa-
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1660 <MARG><I>Tab.</I> 5. <I>fig.</I> 2.</MARG>
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1661 rallela deducatur, qu&aelig; ad $ecundam diametrum per-
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1662 tingat; Quadrilineum comprehen$um ab ip$a curua hy-
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1663 perbolica. &amp; dictis tribus rectis, erit imago velocitatis il-
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1664 <pb n=48>
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1665 lius motus de$cribentis curuam parabolicam, cuius ba$is
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1666 ad axem eius habet eandem rationem, quam duplus axis
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1667 propo$it&aelig; hyperbol&aelig; ad ductam illam &aelig;quidi$t&atilde;tem inter
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1668 eiu$dem hyperbol&aelig; a$$ymptotos interiectam.
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1669 <p>Hyperbol&aelig; IRS $it centrum H, $emiaxis HI, a$$ymptoti
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1670 HT, NH, et SN parallela HI; t&ugrave;m ducta HM $ecunda dia-
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diff changeset
1671 metro hyperbol&aelig;, intelligatur de$criptio parabol&aelig; AFD;
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1672 itaut duplus axis hyperbol&aelig;, hoc e$t quadruplum ip$ius
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1673 HI ad NT eandem habeat rationem, quam DB ba$is pa-
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1674 rabol&aelig; ad BA axim eiu$dem. Dico quadrilineum HISM
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1675 e$$e imaginem velocitatum, iuxta quam motu compo$ito
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1676 de$cribitur parabola AFD; &amp; cum $it homogenea imagi-
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1677 <MARG><I>Def.</I> 7. <I>primi
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1678 &amp; pr.</I> 12. <I>pri-
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1679 mi huius.</I></MARG>
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1680 nibus HILM, HTM, e$$e quoque rectangulum HDLM ad
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1681 imaginem ip$am HISM vt recta CA ad curuam AFD.
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1682 Fiat rectangulum ACDB, et HM $it tempus, quo curritur
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1683 <MARG><I>Cor. pr.</I> 4. <I>hu.</I></MARG>
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1684 vtrunque latus AB, AC, nempe axis AB motu grauium
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1685 iuxta imaginem triangulum HTM, alterum ver&ograve; latus AC
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1686 <MARG><I>Pr.</I> 2. <I>primi
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1687 huius.</I></MARG>
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1688 &aelig;quabili motu iuxta imaginem rectangulum HILM, quod
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1689 quidem erit HILM; etenim AB ad $patium AC e$t vt ima-
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1690 go triangulum HMT ad imaginem rectangulum HILM,
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1691 $cilicet e$t vt MT ad duplam HI, vel vt NT ad quadru-
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1692 plam HI, quemadmodum po$uimus. Iam mon$trauimus
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1693 lineam, qu&aelig; curritur iuxta illas imagines motu compo$ito
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1694 parabolam e$$e, cuius diameter AB, &amp; ba$is BD; &amp; pro-
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1695 pterea erit ip$a AFD (nam vnica tantum parabola ex
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1696 datis AB, BD po$itione, ac magnitudine, axi $cilicet, ac
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1697 ba$i dari pote$t) Ducatur nunc &agrave; quolibet puncto F dict&aelig;
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1698 parabol&aelig; rect&aelig; FE, FG parallelogrammum con$tituentes
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1699 AEFG; &amp; P $it momentum, quo mobile punctum inueni-
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1700 <MARG><I>Ex pr.</I> 12. <I>hu.</I></MARG>
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1701 tur in F. Habebit inibi ip$o temporis momento P veloci-
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1702 tatem PQ iuxta directionem GF, $unt ver&ograve; i$t&aelig; directiones
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1703 $ibi ip$is perpendiculares; ergo recta, qu&aelig; diameter e$$et
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1704 rectanguli AEFG, &amp; ob id potenti&acirc; &aelig;qualis duabus PK,
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1705 <MARG><I>Pr.</I> 3. <I>huius.</I></MARG>
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1706 PQ erit gradus velocitatis, quem mobile habet momen-
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1707 <pb n=49>
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1708 to F motu compo$ito currens; ver&ugrave;m quia quadratum ex
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1709 PR &ecedil;quatur rectangulo ORQ vn&agrave; cum quadrato ex PQ, &amp;
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1710 <MARG><I>Pr.</I> 11. <I>l.</I> 2. <I>co-
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1711 nic.</I></MARG>
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1712 e$t ob hyperbolam rectangulum ORQ &aelig;quale quadrato
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1713 ex HI, vel PK; ergo PR quadratum &aelig;quale erit duobus $i-
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1714 mul quadratis PQ, PK; itaque PR erit gradus velocitatis
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1715 pr&aelig;dicti mobilis in F momento P, compo$itoque motu
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1716 currentis iuxta curuam parabolicam. Pariter momento
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1717 M, cum mobile e$$et in D velocitas compo$iti motus foret
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1718 MS pote$tate &aelig;qualis duabus MT, ML, ac demum in A
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1719 initio motus velocitas e$t HI: quare HISM erit imago ve-
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1720 locitatis motus compo$iti dum mobile punctum de$crip$e-
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1721 <MARG><I>Def.</I> 3. <I>prima
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1722 huius.</I></MARG>
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1723 rit curuam parabolicam AFD, e$tque illa imago imagini-
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1724 bus diui$orum, $eu $implicium, motuum homogenea; ergo
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1725 con$tat ba$im etiam BD ad parabolam AFD eandem ha-
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1726 bere rationem, quam rectangulum HILM ad quadrili-
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1727 neum HISM. Quod &amp;c.
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1728 <C><I>Corollarium. I.</I></C>
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1729 <p><I>Patet, cum latera compo$iti motus $int duo, &amp; $ibi ip$is per-
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1730 pendicularia, tunc gradum velocitatis e&igrave;u$dem motus compo-
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1731 $iti &aelig;qualem e$$e potenti&acirc; duobus $imul gradibus, quos habet
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1732 mobile eodem momento, ac $i $eorfim intelligatur in ip$is ferri
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1733 lateribus.</I>
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1734 <C><I>Corollarium. II.</I></C>
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1735 <p><I>Si ver&ograve; con$iderentur imagines primi $ecundique Ca$us
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1736 inter$e homogenea, erit vt quadrilineum HISM primi ad</I>
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1737 <MARG><I>Pr</I> 2. <I>prim&atail;
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1738 huius.</I></MARG>
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1739 <I>quadrilineum ij$dem literis notatum $ecundi ca$us, vt cur-
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1740 ua illa parabolica ad hanc $ecundi ca$us parabolam.</I>
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1741 <pb n=50>
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1742 <C><I>Corollarium. III.</I></C>
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1743 <p><I>Illud etiam con$tat, e$$e in vtroque ca$u vt quadrilineum
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1744 HIRP ad ip$um PRSM, ita AF ad FD.</I>
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1745 <C>PROP. XIV. THEOR. X.</C>
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1746 <p>PRopo$itis Spirali Archimedea prim&aelig; circulationis
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1747 <MARG><I>Tab.</I> 5. <I>fig.</I> 3.</MARG>
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1748 ABD, et AGF c&otilde;muni parabola, $it FG ba$is huius
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1749 &aelig;qualis radio DA, et GA $it dimidium circumferenti&ecedil; cir-
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1750 culi AEG; erit parabola AGF axem habens GA &aelig;qualis
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1751 propo$it&aelig; $pirali.
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1752 <p>Sit PNK communis hyperbola, cuius coniugati $emia-
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1753 <MARG><I>Pr.</I> 13. <I>huius.</I></MARG>
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1754 xes $int IK, IH, &amp; a$$ymptotos IO. E$to etiam axis hy-
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1755 perbol&aelig; huius, dupla $cilicet IK, ad HO illi &ecedil;quidi$tantem
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1756 vt FG ad AG. Iam con$tat quadrilineum IHPK fore ima-
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1757 ginem velocitatum, iuxta quam curreretur parabola AGF
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1758 tempore IH: $i modo o$tendimus hoc ip$um quadriline&utilde;
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1759 e$$e pariter homogeneam imaginem alterius compo$iti
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1760 motus, quo videlicet de$cribitur $piralis propo$ita ABD,
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1761 <MARG><I>Pr. <*>. prima.</I></MARG>
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1762 palam erit, ip$am parabolam eidem illi $pirali &aelig;qualem fu-
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1763 turam. Ducatur recta KL, qu&aelig; &aelig;quidi$tet IH; item ex
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1764 quouis puncto Q t&etilde;poris IH alia deducatur recta QRMN
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1765 parallela IK: erit parallelogrammum rectangulum HIKL
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1766 imago velocitatum, iuxta quam curritur FG, et HIO trian-
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1767 gulum imago, qua curritur AG motu grauium de$cenden-
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1768 tium: Ver&ugrave;m quia eodem tempore IH, $i mobile currat
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1769 &aelig;quabili motu DA &aelig;qualem FG, e$t eius imago idem re-
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1770 ctangulum IHKL, curriturque illo eodem tempore IH ($pi-
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1771 rali exigente) omnis circuli circunferentia AGEA &aelig;qua-
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1772 bili etiam motu ab extremitate A radij AD circumducti in
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1773 de$criptione $piralis; ob idque factum e$t, vt IK ad HO e$-
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1774 $et vt DA ad circunferentiam ip$am AGEA; nam hoc mo-
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1775 <pb n=51>
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1776 do rectangulum IH in HO e$t imago velocitatum eiu$-
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1777 dem motus per AGEA. Ducatur nunc ex quocun-
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1778 <MARG><I>Pr.</I> 2. <I>prima.</I></MARG>
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1779 que momento Qlinea QRMN ip$i IK &aelig;quidi$tans, &amp; au-
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1780 $picato motu ex centro D momento I, vt nempe oriatur
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1781 $piralis, intelligatur momento Qventum e$$e in B, quamo-
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1782 brem duct&acirc; DBE, erit rectangulum, $eu imago QIKR ad
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1783 imaginem rectangulum HIKL, ita DB ad DE, in qua ra-
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1784 tione, cum propter $piralem, $it etiam circunferentia AGE
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1785 ad circunferentiam AGEA, erit rectangulum IQ in HO
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1786 imago velocitatis per AGE, e$tque velocitas iuxta tangen-
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1787 tem in E ad velocitatem iuxta tangentem circulum BC in
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1788 B vt ED ad DB, $eu vt HO ad QM; ergo cum iuxta tang&etilde;-
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1789 tem in A, hoc e$t in E velocitas $it HO, erit $ecund&ugrave;m tan-
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1790 gentem circulum BC in B, ip$a QM velocitas; propterea-
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1791 que imago triangulum HIO, qu&aelig; in parabol&aelig; de$criptio-
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1792 ne erat per AG, nunc erit per omnes tangentes circulos $u-
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1793 binde crc$centes ex D in E: $cilicet momento I, erit mobi-
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1794 li puncto $ecund&ugrave;m DA, velocitas IK; momento Q du&mtail;
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1795 ade$t in B, erit $ecund&ugrave;m BE velocitas QR, &amp; iuxta tang&etilde;-
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1796 tem in B circuli BC velocitas QM; qu&aelig; amb&aelig;, hoc e$t ve-
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1797 locitates QR, QM cum $int normaliter direct&aelig;, erit eidem
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1798 <MARG><I>Pr.</I> 8. <I>huius &amp;
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1799 Cor. pr.</I> 13.</MARG>
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1800 mobili in B iuxta $piralem velocitas QN potentia ip$is am-
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1801 babus &aelig;qualis. Similiterque momento H cum mobil&etail;
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1802 fuerit in A, erit velocitas iuxta $piralem, ip$a HP &aelig;qualis
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1803 potenti&acirc; duabus velocitatibus HL iuxta radium, et HO
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1804 iuxta tangentem; &amp; $ic omnino liquet, ip$um quadrilineum
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1805 HIKP e$$e imaginem velocitatum tam in de$criptione pa-
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1806 rabol&aelig; AGF, qu&agrave;m $piralis Archimede&aelig; DBA, &amp; cum $it
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1807 in ij$dem de$criptionibus homogenea $ibi ip$i, con$tat ip-
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1808 <MARG><I>Pr.</I> 2. <I>huius.</I></MARG>
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1809 $as curuis &aelig;quales e$$e. Nam vt imago illa ad $e ip$am ita
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1810 parabola ad $piralem pr&aelig;dictam. Quod &amp;c.
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1811 <pb n=52>
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1812 <C><I>Corollarium.</I></C>
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1813 <p><I>Hinc aparet, $piralem DB ad $piralem DBG eandem habe-
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1814 re rationem, quam quadrilineum QIKN ad quadrilineum
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1815 HIKP; pariterque rectam DA ad eandem $piralem DCB ha-
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1816 bere ip$am rationem, ac rectangulum HIKL ad dictum qua-
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1817 drilineum HIKP. Eodem fer&egrave; modo exhiberi pi&szlig;et ratio $pi-
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1818 ralis ad $piralem, lic&egrave;t plurium inter$e circulationum, eritque
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1819 pror$us ea, quam habet vnum ad alterum eiu$dem illius na-
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1820 tur&aelig;, quadrilineorum.</I>
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1821 <C>PROP. XV. THEOR. XI.</C>
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1822 <MARG><I>Tab.</I> 5. <I>Fig.</I> 4.</MARG>
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1823 <p>SPiralis orta ex motu naturaliter accelerato per radi&utilde;
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1824 circuli comprehendentis $piralem ip$am, &amp; ex motu
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1825 &aelig;quabili circa circumferenti&atilde; eiu$dem circuli, &aelig;qualis e$t
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1826 ei curu&aelig; parabolic&aelig; nat&aelig; ex motu compo$ito, cuius vnum
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1827 latus curritur iuxta imaginem trianguli, nempe motu gra-
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1828 uium, alterum ver&ograve; latus iuxta imaginem trilinei $ecundi,
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1829 habebitque parabola ip$a axim &aelig;qualem radio, &amp; ba$i&mtail;
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1830 terti&aelig; parti circunferenti&aelig; eiu$dem circuli $piralem com-
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1831 prehendentis.
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1832 <p>E$to $piralis ACB, qu&aelig; $ignatur ex motu p&utilde;cti A &aelig;qua
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1833 biliter lati circa circumferentiam ADA, dum nempe eod&etilde;
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diff changeset
1834 tempore IF, punctum B currit &agrave; quiete lineam BA motu
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diff changeset
1835 grauium de$cendentium; $it ver&ograve; imago velocitatum dicti
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diff changeset
1836 motus &aelig;quabilis per ADA rectangulum HGFI, &amp; alte-
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1837 <MARG><I>Cor. pr.</I> 4.
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1838 <I>huius.</I></MARG>
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1839 rius motus imago, (qu&aelig; triangulum erit) e$to FEIM. Pa-
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1840 <MARG><I>Pr.</I> 2. <I>prim&atail;</I></MARG>
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1841 tet, quia ip$&aelig; imagines ponuntur homogene&aelig;, e$$e rectan-
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1842 gulum HGFI ad triangulum IFM vt ADA circumferentia
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1843 ad radium BA, &amp; propterea IM ad IH erit vt BA ad dimi-
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diff changeset
1844 dium circunferenti&aelig; AEDA. Sumatur quodlibet mom&etilde;-
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1845 tum K, &amp; ducatur ONKL &aelig;quidi$tans HM, puteturque
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1846 <pb n=53>
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1847 eodem illo momento mobile v&etilde;tum e$$e in C $piralis pro-
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1848 po$it&aelig; BCA: agatur per ip$um punctum radius BCD, &amp; $ic
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1849 illo momento extremitas A currendo circa periph&aelig;riam
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1850 reperietur in D, eritque circonferentia AED ad ip$am
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1851 AEDA, vt imago rectangulum OGFK ad imagin&etilde; GHIF,
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1852 hoc e$t erit vt KF ad FI; at BC ad BD erit vt imago trian-
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1853 gulum KFL ad triangulum FIM, nempe vt quadratum KF
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1854 ad quadratum FI, e$t autem vt BD ad BC ita velocitas
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1855 iuxta tangentem in D ad velocitatem iuxta tangentem in
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1856 C circulum, cuius radius BC; $cilicet ita velocitas IH ad
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1857 velocitatem KN, quadrati nempe IF ad quadratum KF, &amp;
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1858 ob id velocitates, qu&aelig; $unt iuxta tangentes circulos $ubin-
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1859 de cre$c&etilde;tes ex centro B, er&utilde;t expre$$&aelig; in trilineo HNFIH
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1860 $ecundo, cuius $cilicet indoles e$t vt ab$ci$$arum quadrata
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1861 $int vt applicat&aelig;. His compo$itis, intellecti$que erit in B,
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1862 momento F, nulla velocitas, in C momento K du&aelig; velo-
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1863 citate<*> quarum vn&agrave; KI mobile iret iuxta CD, $ed cum al-
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1864 tera $it KN iuxta tangentem circulum, cuius radius CB, ne-
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1865 <MARG><I>Pr.</I> 8. <I>huius.</I></MARG>
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1866 ctitur vna ex duabus illis, quibus ei$d&etilde; potentia e$t &aelig;qua-
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1867 <MARG><I>Cor. prop.</I> 13.
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1868 <I>huius.</I></MARG>
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1869 lis, &amp; qua idem mobile mouetur iuxta $piralem illo mo-
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1870 mento K. Similiter cum mobile e$t in D, $cilicet momento
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1871 I, habebit velocitatem potentia &aelig;qualem HI, qua dirigitur
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1872 iuxta tangentem, &amp; velocitati IM, qua $ecund&ugrave;m radium,
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1873 Itaque imago velocitatum mobilis de$cribentis $piralem
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1874 propo$itis motibus tempore IF, ea erit, cuius applicat&aelig;
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1875 $unt vbique &aelig;quales potentia ijs applicatis, qu&aelig; ab eod&etilde;
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1876 momento intelligi queunt in imaginibus $implicibus, nem-
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1877 pe partialium motuum, HNFI, IFM. Cum pr&aelig;terea OT
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1878 ponatur tertia pars e$$e circumferenti&aelig; AEDA, &amp; e$t eti&atilde;
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1879 trilineum HFI vtpote $ecundum tertia pars parallelogr&atilde;-
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1880 mi HGFI, erit triangulum IFM ad trilineum ip$um HFI vt
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1881 <MARG><I>Pr.</I> 10. <I>primi
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1882 huius.</I></MARG>
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1883 BA, vel ei &aelig;qualis QO ad OT; curritur ver&ograve; vt $upponi-
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1884 tur OQ tempore IF iuxta imaginem triangulum IFM, ergo
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1885 <MARG><I>Pr.</I> 2. <I>primi
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1886 huius.</I></MARG>
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1887 eodem tempore iuxta trilineum HNF curretur alterum la-
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1888 <pb n=54>
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1889 tus OT, $iue ba$is parabol&aelig; QI. Si itaque parabola ip$a
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1890 putetur e$$e ORI, in qua punctum R e$to vbi mobile ade$t
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1891 momento K, deducantur ver&ograve; ab eodem illo puncto RS
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1892 parallela axi QO, et RP &aelig;quidi$tans QI, vel OT, profect&ograve;
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1893 in O, momento F, $icuti in $pirali, nulla erit mobili veloci-
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1894 tas, $ed cum e$t in R momento K habebit geminam veloci-
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1895 tatem, KL $ecund&ugrave;m SR, et KN iuxta PR perpendicularem
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1896 ip$i SR, qu&aelig; du&aelig; velocitates itidem component vnicam
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1897 potentia $imul illis &aelig;qualem, &amp; cum idem dicatur de qui-
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1898 bu$cunque alijs punctis parabol&aelig;, momentis temporis FI
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1899 re$pondentibus, manife$tum e$t $pirali BCA, &amp; parabol&aelig;
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1900 ORI vnicam, eandemque e$$e imaginem velocitatum, pro-
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1901 pterquam qu&ograve;d ip$&aelig; curu&aelig;, qu&ograve;d $int vt imagines, erunt
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1902 inter$e &aelig;quales.
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1903 <MARG><I>Pr.</I> 2. <I>prima.</I></MARG>
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1904 <C><I>Scholium.</I></C>
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1905 <p><I>Exemplo traditarum curuarum, po$$unt innumer&aelig; $pira-
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1906 les $uis parabolis &aelig;quales excogitari, nec ideo res min&ugrave;s de-
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1907 mon$trabitur, $i loco rectarum, $eu laterum OT, OP compo$iti
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1908 motus, $ub$tituantur circuli, aut circulorum arcus, qui ad re-
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diff changeset
1909 ctos angulos $e $ecent, $cilicet c&utilde; tangentes ad punctum infle-
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parents:
diff changeset
1910 xionis, $eu occur$us ip$arum curuarum $ibi ip$is perpendicu-
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diff changeset
1911 res fuerint. Qu&ograve;d $i ip$a curua latera ad rectos angulos non
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1912 $e $ecent curu&aelig; nihilominus ab ip$o compo$ito motu na$cen-
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1913 tes poterunt exhiberi curuas parabolicas exequantes, quarum
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1914 itidem latera $int rect&aelig; eundem angulum, quem pradict&aelig; t&atilde;-
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parents:
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1915 gentes, comprehendentes. Sed de his $atis, nunc dicamus ea
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parents:
diff changeset
1916 tempora, quibus duorum pendulorum $imiles vibrationes ab-
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1917 $oluuntur, hoc e$t Galilei $ententiam demon$trabimus, quam
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1918 quondam haud ruditer decepti fal$am credidimus.</I>
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parents:
diff changeset
1919 <p><I>Vincentius Viuianus eximius no$tri &aelig;ui Geometra vt tue-
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parents:
diff changeset
1920 retur Galilei $ententiam, cuius digni$$im&egrave; $e fui$$e di$<*>ipu-
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diff changeset
1921 lum profitetur, tradidit mihi per admodum Reuerendum, at-</I>
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diff changeset
1922 <pb n=55>
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diff changeset
1923 <I>que culti$$imum Patrem Io$eph Ferronum &egrave; Societate Ie$u, de-
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parents:
diff changeset
1924 mon$trationem $uam ver&egrave; pulcherrimam, ac di$erti$$im&egrave;
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parents:
diff changeset
1925 exaratam, qua vna potui$$em de Galilei a&szlig;erto $atisfactus
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parents:
diff changeset
1926 e$$e; eam demon$trationem, ij$dem pror$us verbis, ac figuris,
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parents:
diff changeset
1927 quibus ad me peruenit hic duxi reponendam, ne gloria&mtail;,
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parents:
diff changeset
1928 quam Vir tantus meretur, ip$i videremur no$tra, quam inde
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parents:
diff changeset
1929 $ubdemus, demon$tratione, $ubripere.</I>
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parents:
diff changeset
1930 <p><I>Inquit ergo.</I>
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parents:
diff changeset
1931 <p>TEmpora naturalium de cur$uum $ph&aelig;rarum grauium
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diff changeset
1932 <MARG><I>Tab.</I> 6. <I>fig.</I> 1. 2
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parents:
diff changeset
1933 3. 4.</MARG>
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parents:
diff changeset
1934 per $imiles, $imiliterque ad horizontem inclinatos
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parents:
diff changeset
1935 arcus curuarum linearum in planis, aut verticalibus, aut
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parents:
diff changeset
1936 ad horizontem &aelig;qualiter inclinatis de$criptarum, &amp; qu&aelig;
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parents:
diff changeset
1937 tot&aelig; $int ad ea$dem partes cau&aelig;, inter$e $unt in $ubdupli-
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parents:
diff changeset
1938 cata ratione chordarum eorundem arcuum homolog&egrave;
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parents:
diff changeset
1939 $umptarum.
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parents:
diff changeset
1940 <p>Ex puncto A ad curuam lineam BCD extra ip$am i&ntail;
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parents:
diff changeset
1941 plano po$itam, &amp; in totum ad ea$dem partes cauam, qu&aelig;-
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parents:
diff changeset
1942 cunque ea $it (vel nimirum pars aliqua circumferenti&aelig;
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parents:
diff changeset
1943 circuli, vel alicuius ex infinitis ellip$ibus, aut parabolis, aut
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parents:
diff changeset
1944 hyperbolis, aut $piralibus, aut cycloidibus, vel concoidis,
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parents:
diff changeset
1945 vel ci$oidis, $eu alterius cuiu$cumque ex notis, vel ignotis
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parents:
diff changeset
1946 curuis educantur omnes rect&aelig; AB, AC, AD &amp;c. qu&aelig; &agrave;
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parents:
diff changeset
1947 punctis E, F, C, vel intra, vel extra eas $umptis proportio-
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parents:
diff changeset
1948 nalibus $ecentur, ita vt $it AB ad AE, $icut AC ad AF, &amp;
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parents:
diff changeset
1949 $icut AD ad AG &amp;c. &amp; hoc $emper. Sic enim dubio pro-
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parents:
diff changeset
1950 cul apparet, prout facillimum e$t o$tendere, lineam EFG
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parents:
diff changeset
1951 tran$euntem per $ingula puncta E, F, G $ic inuenta, cur-
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parents:
diff changeset
1952 uam quoq; e$$e, &amp; eiu$dem penitus natur&aelig;, ac data BCD
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parents:
diff changeset
1953 eique $imilem, $imiliterque cum ip$a po$itam, atque in to-
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parents:
diff changeset
1954 tum cauam ad ea$dem partes, ad quas ponitur caua ip$&atail;
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parents:
diff changeset
1955 BCD. Concipiatur mod&ograve; planum, in quo manent huiu$-
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parents:
diff changeset
1956 modi $imilium curuarum $imiles arcus BCD, EFG, vel e$$e
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parents:
diff changeset
1957 ad horizontem erectum, nemp&egrave; verticale, vel ad ip$u&mtail;
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parents:
diff changeset
1958 <pb n=56>
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parents:
diff changeset
1959 horizontem inclinatum iuxta curuitates ip$orum arcuum
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parents:
diff changeset
1960 BCD, EFG inflexas e$$e $uperficies eidem plano erectas,
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parents:
diff changeset
1961 ita tamen, vt $uper has po$itis grauibus $ph&aelig;ris in A, E per
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parents:
diff changeset
1962 ip$as $ic inflexas $uperficies e&aelig;dem $ph&aelig;r&aelig; naturaliter
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parents:
diff changeset
1963 decurrere queant; id quod $an&egrave; accidet, cum arcus BCD
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parents:
diff changeset
1964 totus fuerit infra horizontalem IL ex arcus $ubli-
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parents:
diff changeset
1965 miori puncto B ductam, fuerintque ab hac continuati re-
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parents:
diff changeset
1966 ce$$us, ac totus ad vnam partem perpendiculi BH: nam $ic
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parents:
diff changeset
1967 talis quoque erit alter arcus EFG illi BCD $imilis, $imili-
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casties
parents:
diff changeset
1968 terque po$itus. His omnibus $ic manentibus: Dico tem-
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parents:
diff changeset
1969 pus decur$us $ph&aelig;r&aelig; grauis E per $imilem, $imiliterque po-
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parents:
diff changeset
1970 $itum arcum EFG, e$$e in $ubduplicata ratione chordarum
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parents:
diff changeset
1971 BO, EG arcus ip$os $ubtendentium. Secto enim b<*>fariam
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parents:
diff changeset
1972 angulo BAD per rectam AC arcum BD $ecantem in C,
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parents:
diff changeset
1973 atque arcum EFG in F, iungantur chord&aelig; BC, CD, et EF,
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parents:
diff changeset
1974 FG, qu&aelig; ex huiu$modi curuarum natura cadent tot&aelig; intra
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parents:
diff changeset
1975 ip$os arcus, $ed in prima, &amp; $ecunda figura ad partes poli
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parents:
diff changeset
1976 A, in tertia ver&ograve;, &amp; quarta ad oppo$itas.
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parents:
diff changeset
1977 <p>Et quoniam, ex talium curuarum gene$i, e$t vt BA ad
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parents:
diff changeset
1978 AE, ita DA, ad AG, erit BD ip$i EG parallela, hoc e$t
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parents:
diff changeset
1979 vtraque ad horizontem &aelig;qualiter inclinata, atque in ra-
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casties
parents:
diff changeset
1980 tione BA ad AE. Similiter cum $it, vt BA ad AE, ita CA
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parents:
diff changeset
1981 ad AF, etiam BC, EF inter$e &aelig;quidi$tabunt, $eu ad hori-
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parents:
diff changeset
1982 zontem &aelig;qualiter inclinabuntur, eruntque in ratione ea-
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parents:
diff changeset
1983 dem, ac BA ad AE. Idemque o$tenditur de chordis CD,
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parents:
diff changeset
1984 FG, quare ex magni Galilei $ententia de motu naturaliter
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parents:
diff changeset
1985 accelerato indubitanter $equitur tempus decur$us $ph&aelig;r&aelig;
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parents:
diff changeset
1986 grauis ex B in D per binas chordas BC, CD ad tempus
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parents:
diff changeset
1987 decur$us per vnicam BD, e$$e vt tempus decur$us grauis
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parents:
diff changeset
1988 $ph&aelig;r&aelig; ex E in G per binas EF, FG ad tempus decur$us
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parents:
diff changeset
1989 per vnicam EG: cadem itidem ratione demon$tratur (an-
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parents:
diff changeset
1990 gulis pariter BAC, CAD bifariam $ectis per rectas, qu&aelig;
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parents:
diff changeset
1991 $imiles arcus BC, EF, ac CD, FG duas in partes diuidant)
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parents:
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1992 ex quatuor vtrinque arcuum horum cordis, illas inter$e
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parents:
diff changeset
1993 <pb n=57>
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casties
parents:
diff changeset
1994 homologas, $imile$que arcus $ubtendentes ad horizonte m
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parents:
diff changeset
1995 e$$e &aelig;qualiter inclinatas, ac alteram alteri in ratione ea-
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parents:
diff changeset
1996 dem, in qua $unt rect&aelig; AB, AE &amp;c: ac propterea ex ea-
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parents:
diff changeset
1997 dem Galilei $cientia con$tabit vtique, tempus decur$us ex
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parents:
diff changeset
1998 B in C $ph&aelig;r&aelig; grauis B per quatuor chordas quatuor par-
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parents:
diff changeset
1999 tes arcus BCD $ubtendentes ad tempus decur$us per vni-
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parents:
diff changeset
2000 cam BD, e$$e vt tempus decur$us $ph&aelig;r&aelig; grauis E ex E in
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parents:
diff changeset
2001 G per quatuor illis homologas chordas quatuor partes
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parents:
diff changeset
2002 arcus EFG pariter $ubtendentes ad tempus decur$us per
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parents:
diff changeset
2003 vnicam chordam EG: &amp; hoc $emper ita euenire demon-
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parents:
diff changeset
2004 $trabitur quantacunque, &amp; maxima fuerit in perpetua an-
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parents:
diff changeset
2005 gulorum bi$ectione &aelig;qu&egrave;multiplicitas in vtroque arcu
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casties
parents:
diff changeset
2006 talium chordarum homolog&egrave; $umptarum, ac inter$e pro-
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casties
parents:
diff changeset
2007 portionalium, &aelig;qualiterque ad horizontem inclinatarum:
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parents:
diff changeset
2008 Propterquam qu&ograve;d $emper decur$us ex B in D per aggre-
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casties
parents:
diff changeset
2009 gatum chordarum omnium in arcu BCD ad tempus de-
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casties
parents:
diff changeset
2010 cur$us per $olam chordam BD e$$e vt tempus decur$us ex
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casties
parents:
diff changeset
2011 E in G per aggregatum totidem chordarum in arcu EFG
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parents:
diff changeset
2012 ad tempus decur$us per vnicam chordam EG; adeo vt de-
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parents:
diff changeset
2013 nique iure optimo educi po$$e videatur, tempus decur$us
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parents:
diff changeset
2014 grauis ex B in D per aggregatum infinitarum chordarum
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parents:
diff changeset
2015 totum arcum BCD con$tituentium, $eu tempus per ip$um
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parents:
diff changeset
2016 arcum BCD ad tempus decur$us per $olam cordam BD
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parents:
diff changeset
2017 e$$e vt tempus decur$us grauis ex E in G per aggregatum
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parents:
diff changeset
2018 totidem infinitarum chordarum dictis homolog&egrave; propor-
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parents:
diff changeset
2019 tionalium, &aelig;qualiterque $ingul&aelig; $ingulis ad horizonte&mtail;
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parents:
diff changeset
2020 inclinatarum, ac totum arcum EFG conformantium, $iue
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parents:
diff changeset
2021 vt tempus per ip$um arcum EFG per $olam chordam EG.
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parents:
diff changeset
2022 Quocirca permutando, tempus, decur$us $ph&aelig;r&aelig; grauis B
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parents:
diff changeset
2023 per arcum BCD ad tempus decur$us $ph&aelig;r&aelig; grauis E per
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casties
parents:
diff changeset
2024 arcum $imilem, $imiliterque po$itum EG erit vt tempus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2025 decur$us per chordam BD ad tempus decur$us per chor-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2026 dam EG; $ed ex eadem Galilaica $cientia de motu, tempus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2027 decur$us per chordam BD ad tempus decur$us per &aelig;qua-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2028 <pb n=58>
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casties
parents:
diff changeset
2029 iter inclinatam EG e$t in $ubduplicata ratione ip$aru&mtail;
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casties
parents:
diff changeset
2030 chordarum BD, EG; ergo tempus quoque decur$us ex B
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2031 per arcum BCD ad tempus decur$us ex E per arcum EFG
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casties
parents:
diff changeset
2032 e$t in eadem $ubduplicata ratione chord&ecedil; BD ad chordam
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2033 EG, quod o$tendendum propo$uimus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2034 <C><I>Corollarium.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2035 <p><I>Ex mod&ograve; <*>sten$is $uper prima, ac $ecunda figura, manife-
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casties
parents:
diff changeset
2036 $tum fit celeberrimum illud magni Galilei pronuntiatum,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2037 qu&ograve;d videlicet, ratio temporum $imilium vibrationum pen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2038 dulorum $it $ubduplicata rationis longitudinum filorum ho-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2039 molog&egrave; $umptorum, non tantum verum e$$e de vibrationibus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2040 pendulorum per arcus $imiles, $imiliterque po$itos, $umptos
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2041 ex circulorum quadrantibus ad perpendiculum v$que termi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2042 nantes, $ed etiam de vibrationibus per arcus quo$cumque $i-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2043 miles quadrantum &agrave; perpendiculo $eiunctos: dummodo ip$i
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2044 $imiles arcus $int quoque $imiliter po$iti: quales nimir&ugrave;m ap-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2045 parent in figuris prima, ac $ecunda arcus BCD, EFG, dum
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casties
parents:
diff changeset
2046 grauia B, E ex filis, aut ha$tulis AB, AE circa punctum A
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2047 conuertibilibus appen$a concipiantur.</I>
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casties
parents:
diff changeset
2048 <C><I>Scholium.</I></C>
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casties
parents:
diff changeset
2049 <p><I>Si curua BCD, EFG in prima, &amp; $ecunda figura fuerint
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casties
parents:
diff changeset
2050 $imiles arcus ex circulis commune centrum A habentibus; ac
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casties
parents:
diff changeset
2051 in verticali plano po$itis, &amp; in prima figura recta AB, AE
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2052 fuerint fila aut ha$tul&aelig; qu&aelig;dam circa clauum A conuertibi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2053 les, in $ecunda ver&ograve; recta AB, AE concipiantur, vt ha$tul&aelig;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2054 inflexibiles, volubile$que circa imum punctum E, atque ex
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2055 huiu$<*>odi filorum, aut ha$tularum terminis B, E pendeant
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2056 graues $ph&aelig;r&aelig; B, E (cum eadem $int tempora prout a$$umi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2057 tur quoque ab ip$o met Ceua) tempora inquam decur$uum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2058 liberorum granium B, E per arcus BCD, EFG, ac tempor&atail;</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2059 <pb n=59>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2060 <I>de$cen$uum ip$orum grauium per eo$dem arcus (vel hac &agrave;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2061 filis pendeant, vel ab hastulis $ustineantur) erit quoque tem-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2062 pus de$cen$us, $eu vibrationis penduli B per arcum BCD ad
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2063 tempus de$cen$us, $eu vibrationis penduli E per arcum EFD
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2064 in $ubduplicata ratione chord&aelig; BD ad chordam EG; $ed h&aelig;c
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2065 ratio chordarum BD, EG eadem e$t, ac ratio filorum, aut ha-
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casties
parents:
diff changeset
2066 $tularum AB, AE; Ergo tempus vibrationis penduli AB per
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2067 arcum BCD ad tempus vibrationis penduli AE per arcum il-
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casties
parents:
diff changeset
2068 li $imilem, $imiliterque po$itum EFG est quoque in $ubdupli-
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casties
parents:
diff changeset
2069 cata ratione longitudinum, vel filorum, aut ha$tularum, ex
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casties
parents:
diff changeset
2070 quibus eadem grauia pendula $imiles vibrationes ab$oluunt
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casties
parents:
diff changeset
2071 BCD, EFG.</I>
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casties
parents:
diff changeset
2072 <C><I>Scholium.</I></C>
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parents:
diff changeset
2073 <p><I>C&aelig;ter&ugrave;m non me latet con$tructionem, ac demonstratio-</I>
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casties
parents:
diff changeset
2074 <MARG><I>Tab.</I> 6 <I>fig.</I> 5.</MARG>
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casties
parents:
diff changeset
2075 <I>nem &agrave; nobis $uperi&ugrave;s allatam nonnullis euidentiorem forta$$e
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casties
parents:
diff changeset
2076 eua$uram, $i ommi$$a illa continua bi$ectione angulorum $i-
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casties
parents:
diff changeset
2077 miles, $imiliterque po$itos arcus ab$cindentium ex $imilibus
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casties
parents:
diff changeset
2078 curuis ibidem de$criptis; atque ommi$$a pariter continua co-
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casties
parents:
diff changeset
2079 niunctione chordarum, vt ibi factum fuit, horum vice, vt in
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casties
parents:
diff changeset
2080 quinta figura, ex punctis B, D bin&aelig; tangentes curuam BCD
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casties
parents:
diff changeset
2081 ducantur BH, DH, qu&aelig; omnin&ograve; mutu&ograve; $e $ecabunt in puncto
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casties
parents:
diff changeset
2082 H (ob conditiones in ip$a Theorematis expo$itione vltimo lo-
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casties
parents:
diff changeset
2083 co po$itas) atque ex E, G ip$is BH, DH agantur &aelig;quidistan-
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casties
parents:
diff changeset
2084 tes, qu&aelig; iunct&aelig;, AH $imul occurrent in I, curuamque EFG
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2085 contingent pariter ad E, G (qu&aelig; omnia $i opus fuerit, facil&egrave;
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casties
parents:
diff changeset
2086 demon$trabuntur) ac in$uper, $i &agrave; puncto C, in quo iunct&atail;
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casties
parents:
diff changeset
2087 AH $ecat arcum BCD, agatur tangens LM primas BH, DH
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parents:
diff changeset
2088 $ecans in LM; Per F ver&ograve;, in quo AICH $ecat arcum EFG
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diff changeset
2089 agatur NO parallela tangenti LM, qu&aelig; curuam pariter EFG
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diff changeset
2090 tanget ad F, ac tangentes EI, GI $ecabit ad NO: &amp; $i iunctis
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diff changeset
2091 in$uper AL, AM, eadem, quam nunc explicauimus, continue-
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diff changeset
2092 tur con$tructio per alias, atque alias tangentes, ac parallelas</I>
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diff changeset
2093 <pb n=60>
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parents:
diff changeset
2094 <I>&amp;c. $ic enim vnicuique harum curuarum circum$cribetur
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parents:
diff changeset
2095 rectilineum, prim&ograve; ex binis tangentibus, $ecund&ograve; ex tribus,
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parents:
diff changeset
2096 terti&ograve; ex quinque, quart&ograve; ex $eptem, &amp; $ic vlteri&ugrave;s iuxta re-
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diff changeset
2097 liquos impares numeros $ucce$$iu&egrave; $umptos; atque omnia pa-
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diff changeset
2098 ria talium &aelig;quidi$tantium tangentium eam $emper inter $e
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parents:
diff changeset
2099 rationem $eruabunt, quam habent chorda BD, EG, $en quam
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diff changeset
2100 habent rect&aelig; BA, EA, eruntq; inter$e &aelig;qualiter inclinat&aelig;;
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parents:
diff changeset
2101 adeoq; tempora decur$uum grauium B, E tam per $ummas
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parents:
diff changeset
2102 binarum tangentium BH, HD, EI, IG, qu&agrave;m per minores
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diff changeset
2103 $ummas, ex quinque $imul chordis vtrinque $umptas, aut
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diff changeset
2104 qu&agrave;m per alias $emper minores $ummas huiu$modi tangen-
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parents:
diff changeset
2105 tium iuxta quantumuis maiorem numerum imparem &aelig;qu&egrave;
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diff changeset
2106 multipliciter $umptarum, erunt perpetu&ograve; proportionalia tem-
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diff changeset
2107 poribus decur$uum per chordas BD, EG; &amp; hoc $emper; etiam-
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parents:
diff changeset
2108 $i per huiu$modi decrementa aggregaterum ex tangentibus
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parents:
diff changeset
2109 vtrinque &aelig;qu&egrave;multipliciter $umptis, deueniatur ad vltim<*>s,
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parents:
diff changeset
2110 ac breui$$imas ip$is arcubus circum$criptiones pol<*>gonorum
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parents:
diff changeset
2111 ex lateribus numero innumerabiliter aqu&egrave;multiplicibus, hoc
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diff changeset
2112 e$t ad ip$os $imiles, $imilitorque po$itos arcus BCD, EFG,
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parents:
diff changeset
2113 quorum $ingula homologorum laterum, $eu punctorum paria,
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parents:
diff changeset
2114 vt B, &amp; E; C et F; D, et G &amp;c. haberi po&szlig;unt tanquam <*>
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parents:
diff changeset
2115 paria parallelarum, ac proportionalium tangentium ip$os $i-
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parents:
diff changeset
2116 miles, ac $imiliter po$itos arcus con$tituentia. Quapropter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2117 ratio quoq; temporum decur$uum per ip$os arcus, $imilis erit
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parents:
diff changeset
2118 rationi temporum decur$uum per chordas; $ed horum decur-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2119 $uum ratio $ubdupla e$t rationis inter ip$as chordas. Quare,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2120 &amp; alia hac methodo con$taret propo$itum.</I>
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parents:
diff changeset
2121 <p><I>Hactenus graui$$imus Vir; $upere$t mod&ograve;, vt quemadmo-
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parents:
diff changeset
2122 dum annuimus, veritatem eandem no$tra quoque methodo,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2123 confirmemus, vt ijs, quibus $atis probat demon$tratio allata,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2124 $it nostra, quam afferemus, in experimentum traditarum h&ugrave;c
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2125 v$q; r&eacute;rum; &amp; quibus $ec&ugrave;s acciderit ex aliqua dubitatione,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2126 h&aelig;c per demon$trationes no$tras pror$us, $<*>atimq tollatur.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2127 Illud etiam admoneo, eam rem non tantum me o$ten$urum,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2128 <pb n=61>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2129 <I>vt pulcherrima, vtilimaq; veritas pluribus demon$trationi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2130 bus aperiatur; ver&ugrave;m potius vt ampli$$ima Methodus, qua tum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2131 vtemur, aliorum motuum demon$trandorum in exemplum
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parents:
diff changeset
2132 veniat.</I>
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parents:
diff changeset
2133 <C>PROP. XVI. THEOR. XII.</C>
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parents:
diff changeset
2134 <p>IN eadem recta CD coeant du&aelig; plan&aelig;, inter$eq; $imiles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2135 <MARG><I>Tab.</I> 6. <I>fig.</I> 7.</MARG>
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parents:
diff changeset
2136 ac pror$us &aelig;quales figur&aelig; ADCA, BDCB, &amp; quidem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2137 ita, vt ab eodem puncto M $i ducatur MH parallela CA,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2138 et ML ip$i CB, $it $emper MH &aelig;qualis ML, quemadmo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2139 dum &aelig;quales $unt inter$e CA, CB. Dico ($i concipiatur
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2140 $olidum eius indolis, vt ductis rectis BA, LH cadant i$t&aelig;
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parents:
diff changeset
2141 omnin&ograve; in $olidi i$tius $uperficie; ip$um ver&ograve; $olidum, quod
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2142 $it BADC, $ecetur plano quolibet &aelig;quidi$tante figur&aelig;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2143 BCD) fore, vt $ectio i$ta KFEIK, $it pror$us $imilis, &aelig;qua-
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parents:
diff changeset
2144 li$que alteri contermin&aelig; AEI; $ed opportet, vt palam e$t,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2145 coeuntes ill&aelig; figur&aelig; non in eodem plano reperiantur.
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parents:
diff changeset
2146 <p>Cum duo plana inuicem parallela KIE, BCD $ecent
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parents:
diff changeset
2147 alia duo inter$e item parallela ACB, HML, erunt commu-
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parents:
diff changeset
2148 nes $ectiones, inter$e omnes &aelig;quidi$tantes rect&aelig; line&aelig; KI,
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casties
parents:
diff changeset
2149 GF, ML, CB. Cum ver&ograve; ob naturam $olidi, $ectiones
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parents:
diff changeset
2150 BAC, IHM triangula $int rectilinea, erit vt BC ad CA,
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parents:
diff changeset
2151 ita KI ad IA. Sunt autem priores inter$e &aelig;quales, ergo &amp;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2152 po$trem&aelig; KI, AI inter$e &aelig;quabuntur. Eademque ratione
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2153 $unt &aelig;quales HG, GF: &amp; quoniam ob fimilitudinem figu-
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parents:
diff changeset
2154 rarum angulus BCD &aelig;quatur angulo ACD, &amp; angulus
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parents:
diff changeset
2155 BCD &aelig;qualis angulo KIE (nam etiam CD, IE $unt rect&aelig;
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diff changeset
2156 &aelig;quidi$tantes, cum nempe $int communes $ectiones plani
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parents:
diff changeset
2157 DCA $ecantis duo &aelig;quidi$tantia KIE, BCD) ergo cu&mtail;
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diff changeset
2158 angulus pariter ACD &aelig;quet angulum AIE, erunt anguli
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parents:
diff changeset
2159 KIE, AIE, et FGE, HGF &aelig;quales. Quod &amp;c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2160 <pb n=62>
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parents:
diff changeset
2161 <C>PROP. XVII. THEOR. XIII.</C>
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diff changeset
2162 <p>II$dem manentibus. Dico triangula ACB, LHM e$$&etail;
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diff changeset
2163 $imilia. Sunt enim parallel&aelig; &amp;c. inter$e tam rect&aelig; CB,
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parents:
diff changeset
2164 ML, qu&agrave;m CA, MH; ideo anguli ACB, HML inter$e
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diff changeset
2165 &aelig;quabuntur, &amp; $unt circa eos proportionalia latera, nem.
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parents:
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2166 pe BC ad CA, vt LM, MH; ergo con$tat propo$itum.
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parents:
diff changeset
2167 <C><I>Corollarium.</I></C>
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2168 <C><I>Simul con$tat rectas AB, LH inter$e &aelig;quidi$tare.</I></C>
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diff changeset
2169 <C>PROP. XVIII. THEOR. XIV.</C>
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diff changeset
2170 <p>II$dem vt $upra manentibus, ita tamen vt ACD $it an-
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2171 gulus rectus ($ic enim DC perpendicularis erit duabus
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2172 AC, CB) Dico $olidum huiu$modi ad pri$ma, cuius ba$is
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2173 ABC, &amp; altitudo CD eandem habere rationem, quam $o-
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casties
parents:
diff changeset
2174 lidum rotundum ortum ex rotatione figur&aelig; CAD circ&atail;
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casties
parents:
diff changeset
2175 axem CD ad cylindrum genitum ex conuer$ione rectan-
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diff changeset
2176 guli AC in CD circa eundem axem.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2177 <MARG><I>Tab.</I> 6. <I>Fig.</I> 8.</MARG>
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parents:
diff changeset
2178 <p>Compleatur ip$um pri$ma, &amp; $it quidem AQDPBC,
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diff changeset
2179 quod $ecetur vn&agrave; cum propo$ito $olido per quoduis pla-
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diff changeset
2180 num ba$i ACB &aelig;quidi$tans: fiet in pri$mate $ectio trian-
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parents:
diff changeset
2181 gulum OMN $imile, &aelig;qualeque ip$i ACB, &amp; in altero $o-
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parents:
diff changeset
2182 lido triangulum LHM eidem ACB $imile. Triangulum
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parents:
diff changeset
2183 ACB pri$matis ad tri&atilde;gulum idem $olido propo$ito com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2184 mune, e$t vt circulus radio CA de$criptus ad circulum
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parents:
diff changeset
2185 eundem; Item triangulum NOM $ectio pri$matis e$t ad
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parents:
diff changeset
2186 triangulum LHM $ectionem propo$iti $olidi, vt circulus ex
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parents:
diff changeset
2187 radio MO de$criptus ad circulum radio MH. Cum dein-
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parents:
diff changeset
2188 de idem dicatur de alijs omnibus $ectionibus pri$matis, &amp;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2189 <pb n=63>
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parents:
diff changeset
2190 propo$iti $olidi erunt omnes $imul prim&aelig;, qu&aelig; inter$&etail;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2191 <MARG><I>I<*>mma</I> 18. <I>in
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parents:
diff changeset
2192 libro de dim.
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parents:
diff changeset
2193 parab. Euang.
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diff changeset
2194 T<*>rric<*>l.</I></MARG>
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parents:
diff changeset
2195 &aelig;quales $unt, ad omnes $imul $ecundas vt omnes terti&aelig;,
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parents:
diff changeset
2196 his partibus inter$e &aelig;qualibus, ad omnes quartas; $cilicet
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diff changeset
2197 erunt omnia triangula pri$matis, $eu ip$um pri$ma ad om-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2198 nia triangula propo$iti $olidi, $eu ad ip$um $olidum, vt om-
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diff changeset
2199 nes circuli eius cylindri, qui oritur ex conuer$ione figur&aelig;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2200 ADCA circa axem CD, hoc e$t vt ip$um $olidum rotun-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2201 dum, $eu cylindrus ad omnes $imul circulos $olidi rotundi
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2202 geniti ex rotatione figur&aelig; AHDCA circa axem ips&utilde; CD,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2203 $eu ad ip$um propo$itum $olidum. Quod &amp;c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2204 <C>PROP. XIX. THEOR. XV.</C>
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diff changeset
2205 <p>ET rur$us ip$a manente figura patet, $i ducantur HR,
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diff changeset
2206 LS parallel&aelig; MD, fore non $olum figuram AHDPA,
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diff changeset
2207 $imilem, ac &aelig;qualem BLDQB; ver&ugrave;m etiam APRHA ip$i
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2208 BLSQB: Cum ita $it, aio, eundem cylindrum ad $oli-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2209 dum rotundum genitum, ex volutatione figur&aelig; APD cir-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2210 ca eundem axem CD eandem rationem habere, ac pri$ma
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diff changeset
2211 pr&aelig;dict&utilde;, cuius ba$is ACB, altitudo AP ad $olidum, quod
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2212 $upere$t ex ip$o pri$mate, dempto $olido ACBLDHA.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2213 <p>Nam ex pr&aelig;terita propo$itione nouimus, dictum pri$ma
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2214 ad $olidum eius partem ACBLDHA e$$e vt cylindrus or-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2215 tus ex conuer$ione rectanguli CP circa axem CD ad par-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2216 tem eius rotundum circa axem eundem CD conuer$a fi-
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diff changeset
2217 gura ADC, ergo per conuer$ionem rationis, erit id quod
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diff changeset
2218 propo$uimus.
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parents:
diff changeset
2219 <C>DEF. IV.</C>
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diff changeset
2220 <p>QVodcunque ex dictis propo$itis $olidis vocetur ab
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diff changeset
2221 ea figura, iuxta quam intelligitur ortum. Scilicet
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diff changeset
2222 ACBLDHA dicatur &agrave; figura AHDCA, &amp; alte-
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diff changeset
2223 rum, quod fuit re$iduum pr&aelig;dictum dicatur &agrave; figura AH-
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parents:
diff changeset
2224 DPA.
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parents:
diff changeset
2225 <pb n=64>
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parents:
diff changeset
2226 <C>PROP. XX. THEOR. XVI.</C>
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diff changeset
2227 <p>SI &agrave; quibu$cunque figuris fuerint duo $olida, h&aelig;c inter-
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parents:
diff changeset
2228 <MARG><I>Tab.</I> 6. <I>Fig.</I> 9.</MARG>
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parents:
diff changeset
2229 $e erunt vt $olida alia genita ex conuer$ione illarum
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diff changeset
2230 figurarum circa communem $ectionem $imilium, &aelig;qua-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2231 lium, ac inter$e coeuntium figurarum.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2232 <p>Solidum &agrave; figura ABC $it CAFDBC, &amp; quod e$t &agrave; fi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2233 gura GLH e$to HGILH. Dico illud ad hoc $olidum e$$e
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2234 vt rotundum natum ex conuer$ione figur&aelig; ABC circ&atail;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2235 axem CE ad rotundum ortum ex c&otilde;uer$ione figur&aelig; GLH
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2236 circa axem HL. Opportet tamen angulos ACF, GHI
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2237 &aelig;quales e$$e. Intelligantur pri$mata triangularia, quorum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2238 ba$es ACF, GHI, &amp; altitudines CE, HL; hoc e$t $int ip$a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2239 $olida pri$matica AFCEBD, GIHLMK. Solidum &agrave; figu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2240 <MARG>18. <I>huius.</I></MARG>
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parents:
diff changeset
2241 ra ABC ad pri$ma AFCEBD habet eandem rationem,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2242 quam $olidum rotundum ortum ex conuer$ione $igur&aelig;
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parents:
diff changeset
2243 ABC circa axem CE ad cylindrum natum ex rotatione
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2244 ABEC circa eundem axem CE; hic ver&ograve; cylindrus ad cy-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2245 lindrum alium natum ex rotatione rectanguli GMLH cir-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2246 ca axem HL e$t vt pri$ma, cuius ba$is ACF, altitudineque
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2247 CE ad alterum pri$ma ba$em habens GHI $imilem ip$i CF
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parents:
diff changeset
2248 (nam circa angulos &aelig;quales H, C $unt latera etiam pro-
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parents:
diff changeset
2249 portionalia, nempe &aelig;qualia) &amp; altitudinem HL. Solidum
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parents:
diff changeset
2250 pr&aelig;terea, hoc e$t pri$ma GKHM ad $olidum, quod e$t &agrave;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2251 <MARG><I>Ex eadem.</I></MARG>
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diff changeset
2252 plano GLH habet eandem rationem, ac cylindrus, qui fit
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diff changeset
2253 ex conuer$ione rectanguli HM circa axem HL ad $olidum
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parents:
diff changeset
2254 rotundum ortum ex circumactione figur&aelig; GLH circa ip-
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parents:
diff changeset
2255 $um axem HL, ergo ex &aelig;quali erit $olidum &agrave; figura ABC
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parents:
diff changeset
2256 ad $olidum &agrave; figura GLH, vt rotundum ex rotatione figu-
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parents:
diff changeset
2257 r&aelig; ABC circa axem CE ad rotundum alterum ex conuer-
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parents:
diff changeset
2258 $ione alterius figur&aelig; GLH circa axem HL. Quod &amp;c.
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parents:
diff changeset
2259 <pb n=65>
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parents:
diff changeset
2260 <C>PROP. XXI. THEOR. XVI.</C>
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parents:
diff changeset
2261 <p>PRopo$itis ij$dem $olidis, erunt inter $e, vt momenta fi-
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parents:
diff changeset
2262 gurarum a quibus $unt, qu&aelig; tamen figur&aelig; $u$pen$&aelig;
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parents:
diff changeset
2263 $int ex longitudinibus deductis ab ip$arum grauitatu&mtail;
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parents:
diff changeset
2264 centris v$que ad coeuntium figurarum communes illas $e-
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parents:
diff changeset
2265 ctiones.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2266 <p>Figur&aelig;, &agrave; quibus $unt $olida, ponantur ABC, GLH, c&etilde;-
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2267 <MARG><I>Tab.</I> 6. <I>fig.</I> 10.</MARG>
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2268 tra grauitatum illarum M, N; axes, $iue communes $ectio-
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diff changeset
2269 nes coeuntium binarum inter$e $imilium, ac &aelig;qualium fi-
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diff changeset
2270 gurarum &agrave; quibus dicuntur ip$a $olida; &amp; demum MO, NP
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2271 perpendiculares $int ab ip$is centris ad illas communes $e-
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diff changeset
2272 ctiones deduct&aelig; CE, HL. Dico, $olidum &agrave; plana figur&atail;
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2273 ABC ad $olidum a plana GHL eandem habere rationem,
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2274 ac momentum figur&aelig; ABC pendentis ex MO ad momen-
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2275 <MARG><I>pr.</I> 20. <I>huius.</I></MARG>
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2276 tum alterius figur&aelig; $u$pen$&aelig; ex NP, $unt enim h&aelig;c $oli-
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2277 da inter$e, vt rotunda, quorum genetrices figur&aelig; ABC,
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diff changeset
2278 GLH circa axes CE, HL, huiu$modi ver&ograve; $olida $unt vt
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2279 <MARG><I>Ter. lem.</I> 31.
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2280 <I>in libro di-
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2281 <*>. p<*> ab<*></I></MARG>
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2282 momenta propo$ita; ergo $olidum &agrave; plana figura ABC ad
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diff changeset
2283 $olidum &agrave; plana GLH, erit vt momentum figur&aelig; ABC
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diff changeset
2284 $u$pen$&aelig; ex MO ad momentum GLH pendentis ex NP.
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diff changeset
2285 Quod &amp;c.
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2286 <C><I>Corollarium.</I></C>
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2287 <p><I>Cum ip$a illa momenta nectantur ex rationibus figurarum
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2288 <MARG>Ex mechani-
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2289 cis,</MARG>
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2290 ABC, GLH, &amp; ex longitudinibus, ex quibus pendent ip$&aelig; fi-
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2291 gura (nam habentur vt grauia) ex ij$dem etiam rationibus
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diff changeset
2292 componentur $olida, qua $unt ab ip$is figuris&mdash;</I>
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2293 <pb n=66>
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2294 <C>PROP. XXII. THEOR. XVII.</C>
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2295 <MARG><I>Tab.</I> 7. <I>fig.</I> 1.</MARG>
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2296 <p>IMagines velocitatum, $eu $patia, qu&aelig; curruntur accele-
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2297 ratis motibus, $unt vt $olida ab imaginibus $implicium
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2298 motuum, ex quibus ip$i gignuntur accelerati.
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2299 <p>Sint imagines $implicium motuum ABC, GLH, &amp; $oli-
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2300 da ab ip$is imaginibus (angulis ACQ, GHD $emper re-
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2301 ctis, aut $altem &aelig;qualibus) intelligantur ABCRQ, GLHD.
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2302 Dico, vt $unt inter$e i$ta $olida, $ic e$$e homolog&egrave; $patium
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2303 exactum tempore AC motu accelerato ex $implici motu
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2304 imaginis ABC ad $patium tran$actum tempore GH motu
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2305 item accelerato ex $implici imagine priori homogene&atail;
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2306 GLH: $ecetur $olidum ABCRQ plano &aelig;quidi$tanti QCR,
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2307 <MARG>16. <I>huius.</I></MARG>
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2308 quod faciat in $olido ip$o $ectionem TSVX: erit h&aelig;c figu-
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2309 ra pror$us $imilis, ac &aelig;qualis contermin&aelig; ABVI; quare
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2310 <MARG>4. <I>huius,</I></MARG>
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2311 cum in accelerato motu velocitas, qu&aelig; habetur momen-
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2312 to C ad velocitatem momento S $it vt imago ABC $im-
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2313 <MARG>16. <I>huius.</I></MARG>
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2314 plex ad $egmentum eius ABVS: erit etiam QCR &aelig;qualis
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2315 ABC ad $ectionem $olidi TSVX, qu&aelig; &aelig;quatur ABVS, vt
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2316 illa eadem velocitas momento C mobili inh&aelig;rens ad ve-
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2317 locitatem momento S alterius accelerati motus. E$t au-
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2318 tem $ectio TSVX ad libitum $umpta; ergo $olidum ABC-
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2319 <MARG><I>Def.</I> .3. <I>primi
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2320 huius.</I></MARG>
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2321 QR pote$t $umi merito vt imago velocitatum accelerati
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2322 <MARG><I>&amp; Def.</I> 1. <I>hu-
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2323 ius vn&agrave; cum
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2324 pr.</I> 4. <I>huius.</I></MARG>
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2325 motus, cuius $implex imago ABC: &amp; eodem modo $oli-
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2326 dum alterum vicem geret imaginis velocitatum alterius
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2327 motus ex $implici imagine GLH, itaque erit ob homoge-
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2328 neitatem $patium tran$actum motu accelerato iuxta $im-
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2329 plicem imaginem ABC ad $patium tran$actum motu ac-
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2330 celerato iuxta $implicem imaginem GLH, t&etilde;poribus AC,
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2331 GH, vt $olidum ABCQR ad ALHD,
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2332 <pb n=67>
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2333 <C>PROP. XXII. THEOR. XVIII.</C>
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2334 <p>SInt nunc CE, HL communes $ectiones imaginum $im-
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2335 <MARG><I>Tab.</I> 7. <I>Fig.</I> 2.</MARG>
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2336 plicium ABC, GLH, $i extenderentur cum $ujs &aelig;qua-
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2337 libus, ac $imilibus coeuntibus figuris. E$to pariter M cen-
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2338 trum grauitatis imaginis ABC, et N grauitatis alterius ima-
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2339 ginis GLH; actis dem&ugrave;m MO, NP perpendicularibus ad
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2340 ip$as CE, HL. Dico, $patium accelerati motus ab imagine
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2341 $implici ABC ad $pati&utilde; accelerati alterius motus ab ima-
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2342 gine $implici GLH componi ex ratione imaginis ABC ad
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2343 imaginem GLH, &amp; ex ea perpendicularis MO ad perpen-
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2344 dicularem NP. Cum h&aelig;c ip$a $patia $int o$ten$a, vt $oli-
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2345 <MARG>21. <I>huius.</I></MARG>
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2346 da &agrave; figuris ABC, GLH; h&aelig;c ver&ograve; $unt vt momenta ip$a-
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2347 <MARG>20. <I>huius.</I></MARG>
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2348 rum figurarum $u$pen$arum ex MO, NP. Ergo quemad-
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2349 modum momenta i$ta nectuntur ex rationibus figurarum
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2350 <MARG><I>Ex mechani-
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2351 cis.</I></MARG>
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2352 tanquam magnitudinum ABC ad LGH, &amp; di$tantiarum
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2353 MO ad NP, ita pariter ex his nectentur propo$ita $patia.
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2354 <C><I>Corollarium.</I></C>
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2355 <p><I>Patet communes $ectiones CE, HL e$$e &aelig;quidi$tantes ap-
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2356 plicatis AB, HL, qu&aelig; in imaginibus $umuntur perpendicula-
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2357 res rectis AC, GH. nam HL est recta, in quam coeunt figura</I>
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2358 <MARG><I>Pr</I> 2. <I>prim&atail;
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diff changeset
2359 huius.</I></MARG>
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diff changeset
2360 <I>plan&aelig; $imiles, ac &aelig;quales.</I>
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diff changeset
2361 <C>PROP. XXIV. THEOR. XIX.</C>
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diff changeset
2362 <p>SI imagines $implicium motuum fuerint $imiles, $imili-
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diff changeset
2363 terque $u$pen$&aelig;, imagines velocitatum accelerato-
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diff changeset
2364 rum motuum erunt in triplicata ratione temporum $impli-
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diff changeset
2365 cium motuum, aut in triplicata homologarum, vel extre-
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diff changeset
2366 marum velocitatum eorundem $implicium motuum.
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diff changeset
2367 <p>Cum centra grauitatum $imilium imaginum, $eu figu-
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diff changeset
2368 <MARG><I>Tab.</I> 7. <I>Fig.</I> 3.</MARG>
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diff changeset
2369 <pb n=68>
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diff changeset
2370 rarum, $int puncta in ij$dem figuris $imiliter po$ita, ponun-
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diff changeset
2371 tur ver&ograve; imagines $imiliter $u$pen$&aelig;, ergo $equitur ip$as
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diff changeset
2372 longitudines e$$e vt latera homologa dictarum imaginum,
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diff changeset
2373 $cilicet vt tempus AC ad tempus FG, vel vt extrem&aelig; ve-
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diff changeset
2374 locitates BC ad KE. Quamobrem imagines ip$&aelig;, cum $int
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diff changeset
2375 in duplicata ratione laterum homologorum, $i huic dupli-
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parents:
diff changeset
2376 cat&aelig; addatur alia ratio $imilis rationi longitudinum, fiet
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diff changeset
2377 ratio imaginum velocitatum, $eu $patiorum acceleratorum
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diff changeset
2378 motuum ex $implicibus illis deriuantium triplicata tempo-
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diff changeset
2379 rum, vel extremarum velocitatum $implicium motuum.
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diff changeset
2380 <C>PROP. XXV. THEOR. XX.</C>
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diff changeset
2381 <p>SI ver&ograve; $implices motus extiterint $imiles, &aelig;qualibu$q;
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diff changeset
2382 temporibus ab$oluantur, imagines acceleratorum
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diff changeset
2383 <MARG><I>Tab.</I> 7. <I>fig.</I> 4.</MARG>
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diff changeset
2384 motuum erunt in $ola ratione amplitudinum imaginum
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diff changeset
2385 $implicium.
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diff changeset
2386 <p>Sint imagines $imilium, ac $implicium motuum BAC,
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diff changeset
2387 KFG, quarum grauitatis centra D, H, erunt ex hypothe$i
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diff changeset
2388 <MARG>8 <I>primi huius</I></MARG>
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diff changeset
2389 tempora AC, FG &aelig;qualia; &amp; ideo $patia, $cilicet imagines
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diff changeset
2390 <MARG>2 <I>primi huius</I></MARG>
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diff changeset
2391 velocitatum BAC, KFG habebunt eandem rationem,
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diff changeset
2392 quam $umm&aelig;, aut extrem&aelig; motuum $implicium velocita-
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diff changeset
2393 tes, $cilicet, quam amplitudines imaginum, $eu gene$um:
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diff changeset
2394 $unt ver&ograve; di$tanti&aelig; DE, HI pariter &aelig;quales, quia AC, FG
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diff changeset
2395 <MARG>23. <I>huius.</I></MARG>
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diff changeset
2396 &aelig;quales $unt; ergo cum $patia acceleratorum motuum ne-
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diff changeset
2397 ctantur ex imaginibus $implicium motuum ABC, KFG, &amp;
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diff changeset
2398 ex di$tantijs DE ad HI, liquet ip$a $patia e$$e in vnica, $o-
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parents:
diff changeset
2399 laque ratione amplitudinum BC, KG, aut amplitudinum
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diff changeset
2400 gene$um.
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parents:
diff changeset
2401 <C>PROP. XXVI. THEOR. XX.</C>
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diff changeset
2402 <p>AT$i $implicium, $imiliumque motuum fuerint imagi-
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diff changeset
2403 nes &aelig;qu&egrave; ampl&aelig;, imagines acceleratorum motuum,
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diff changeset
2404 <pb n=69>
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diff changeset
2405 $iue tempora erunt in duplicata ratione temporum i$to-
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diff changeset
2406 rum, vel illorum motuum.
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2407 <p>Amplitudines imaginum $implicium, velocitatumque
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2408 <MARG><I>Tab.</I> 7. <I>fig.</I> 5.</MARG>
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diff changeset
2409 BAC, KFG $unto BC, KG, qu&aelig; &aelig;quales $int. Dico $pa-
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parents:
diff changeset
2410 tia acceleratorum motuum ab illis $implicibus imaginibus
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diff changeset
2411 fore in duplicata ratione temporum AC ad FG (qu&ecedil; $em-
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diff changeset
2412 per in acceleratis ponuntur eadem, ac in $implicibus, nec
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diff changeset
2413 aliter e$$e po$$unt.) Vt FG ad GK, ita $it AC ad CL, &amp;
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parents:
diff changeset
2414 intelligatur LAC imago alterius motus $imilis motui, cuius
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parents:
diff changeset
2415 imago BAC, vel KFG. Facil&egrave; demon$trabitur ip$am fi-
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diff changeset
2416 <MARG><I>Def.</I> 7. <I>primi
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diff changeset
2417 huius.</I></MARG>
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diff changeset
2418 guram LAC $imilem e$$e ip$i KFG, &amp; ad BAC eande&mtail;
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parents:
diff changeset
2419 habere rationem, quam LC ad BC. Cum ergo imago BAC
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diff changeset
2420 ad imaginem KFG componatur ex ratione imaginis BAC
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diff changeset
2421 ad LAC (qu&aelig; $unt vt BC ad CL) &amp; ex ratione imagi-
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parents:
diff changeset
2422 nis ALC ad imaginem KFG, qu&aelig; $unt in ratione compo-
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parents:
diff changeset
2423 $ita LC ad KG, et AC ad FG: priores ver&ograve; du&aelig; rationes
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diff changeset
2424 componunt vnicam &aelig;qualitatis, ergo relinquitur, imagi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2425 nem BAC ad imaginem KFG e$$e vt AC ad FG; $patium
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parents:
diff changeset
2426 ver&ograve; accelerati motus ex $implici imagine BAC ad accele-
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diff changeset
2427 ratum ex $implici KFG nectitur ex ratione imaginum $im-
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diff changeset
2428 <MARG>23. <I>huius.</I></MARG>
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diff changeset
2429 plicium ip$arum, &amp; ex ea di$tantiarum DE, HI &agrave; centris
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diff changeset
2430 grauitatum deductarum D, H, et $unt h&aelig; rect&aelig; in eadem
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parents:
diff changeset
2431 ratione, ac altitudines AC, FG (nam in figuris, $eu imagi-
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diff changeset
2432 nibus $imilium motuum BAC, LAC centra grauitatum
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diff changeset
2433 $unt in eadem recta parallela ip$i BC, &amp; in LAC, KFG
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diff changeset
2434 $unt in punctis $imiliter po$itis, adeout, $icut po$itum e$t,
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diff changeset
2435 ratio ip$arum di$tantiarum in ip$is figuris LAC, KFG, $eu
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parents:
diff changeset
2436 BAC, KEG eadem $it, ac laterum homologorum LC ad
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diff changeset
2437 KG, vel AC ad FG) ergo $patium accelerati motus ex $im-
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diff changeset
2438 plici imagine KFG, erit vt quadratum ex AC ad quadra-
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diff changeset
2439 tum ex FG, nempe in duplicata ratione temporum $impli-
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diff changeset
2440 cium motuum.
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parents:
diff changeset
2441 <pb n=70>
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parents:
diff changeset
2442 <C>PROP. XXVII. THEOR. XXI.</C>
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2443 <MARG><I>Tab.</I> 7. <I>fig.</I> 6.</MARG>
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diff changeset
2444 <p>DEm&ugrave;m $i $int imagines, qu&aelig;cunque velocitatnm $im-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2445 plicium, $imiliumque motuum, imagines accelera-
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casties
parents:
diff changeset
2446 torum motuum, $eu $patia ijs motibus exacta componen-
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parents:
diff changeset
2447 tur ex duplicata temporum ratione, &amp; ex ea amplitudi-
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diff changeset
2448 num, vel applicatarum homologarum earundem imagi-
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diff changeset
2449 num.
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casties
parents:
diff changeset
2450 <p>Imagines $imilium, $impliciumque motuum $int BAC,
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diff changeset
2451 KFG. Dico, imagines acceleratorum motuum ab illis $im-
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diff changeset
2452 plicibus deriuantium habere rationem compo$itam ex du-
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diff changeset
2453 plicata temporum AC ad FG, &amp; amplitudinum imaginum
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diff changeset
2454 dictarum, vel gene$um. Intelligatur alius $imilis motus,
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diff changeset
2455 cuius velocitatum imago $it DFG &aelig;qu&egrave;ampla, ac homo-
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diff changeset
2456 genea ip$i BCA; nimirum $it DG &aelig;qualis BC. Quoniam
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diff changeset
2457 imago accelerati motus ex $implici imagine BA ad imagi-
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diff changeset
2458 nem accelerati ex $implici imagine KFG componitur ex
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diff changeset
2459 ratione imaginis accelerati motus, cuius $implex imago
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diff changeset
2460 BAC ad imaginem accelerati motus ex $implici DFG, &amp;
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diff changeset
2461 ex imagine huius accelerati motus ad accelerati imaginem
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diff changeset
2462 &agrave; $implici KFG; e$t autem prior ratio imaginum, $eu $pa-
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diff changeset
2463 tiorum acceleratis motibus percur$orum ip$a temporum
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diff changeset
2464 <MARG><I>Pr.</I> 26 <I>huius.</I></MARG>
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diff changeset
2465 duplicata AC ad FG, &amp; altera dictarum imaginum, $eu
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diff changeset
2466 $patiorum item acceleratis motibus confectorum, &amp; quo-
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diff changeset
2467 <MARG><*>5. <I>huius.</I></MARG>
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diff changeset
2468 rum $implices imagines $unt DFG, KFG, e$t eadem, ac ra-
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diff changeset
2469 tio amplitudinum DG, $eu BC ad KG. Ergo cum i$t&aelig;
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diff changeset
2470 amplitudines $int e&aelig;dem, ac ill&aelig; gene$um, con$tar propo-
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diff changeset
2471 $itam rationem acceleratorum motuum ex $implicibus
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diff changeset
2472 imaginibus BAC, KFG habere rationem compo$itam ex
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diff changeset
2473 duplicata temporum AC ad FG, &amp; ex ea amplitudinum
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diff changeset
2474 imaginum $implicium BC ad KG, $eu amplitudinum gene-
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diff changeset
2475 $um. Quod &amp;c.
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diff changeset
2476 <pb n=71>
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parents:
diff changeset
2477 <C>PROP. XXVIII. THEOR. XXII.</C>
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parents:
diff changeset
2478 <p>SI gene$es $imilium, $impliciumque motuum fuerint
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diff changeset
2479 &aelig;qu&egrave;ampl&aelig;, imagines acceleratorum motuum erunt
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diff changeset
2480 in duplicata ratione temporum, vel altitudinum ip$arum
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diff changeset
2481 gene$um.
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parents:
diff changeset
2482 <p>Gene$es $imilium, ac $implicium motuum $unto ABC,
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diff changeset
2483 <MARG><I>Tab.</I> 7. <I>Fig.</I> 7.</MARG>
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diff changeset
2484 DEF, quarum amplitudines &aelig;quales $int AC, DF. Dico,
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parents:
diff changeset
2485 imagines, $iue $patia acceleratorum motuum e$$e in dupli-
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diff changeset
2486 cata ratione temporum, vel altitudinum BC ad EF. Cum
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diff changeset
2487 AC, DF $int gradus velocitatum in extremitatibus $impli-
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diff changeset
2488 cium decur$uum, etiam imagines velocitatum, iuxta ip$as
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diff changeset
2489 gene$es, qu&aelig; $int inter$e homogene&aelig;, erunt &aelig;qu&egrave;ampl&aelig;,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2490 &amp; $unt $imilium motuum; ergo imagines acceleratorum
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diff changeset
2491 <MARG>26. <I>huius.</I></MARG>
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diff changeset
2492 motuum, iuxta $implices illas gene$es, aut imagines &aelig;qu&egrave;-
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diff changeset
2493 amplas erunt in duplicata ratione temporum: $unt autem
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diff changeset
2494 imagines velocitatum &aelig;qu&egrave;ampl&aelig;, $imiliumque motuum,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2495 <MARG>26. <I>huius.</I></MARG>
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diff changeset
2496 hoc e$t $patia BC ad EF vt ip$a tempora; ergo $patia acce-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2497 leratorum, propo$itorumque motuum erunt in ratione du-
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diff changeset
2498 plicata altitudinum BC, EF $implicium gene$um, ABC,
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parents:
diff changeset
2499 DEF. Quod &amp;c.
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diff changeset
2500 <C>PROP. XXIX. THEOR. XXIII.</C>
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parents:
diff changeset
2501 <p>SI gene$es $imilium, $impliciumque motuum fuerint
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diff changeset
2502 &aelig;qu&egrave;alt&aelig;, imagines, $iue $patia, acceleratorum mo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2503 tuum erunt vt tempora, vel reciproc&egrave; vt amplitudines ge-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2504 ne$um ip$orum $implicium motuum.
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parents:
diff changeset
2505 <p>Gene$es $imilium, $impliciumque motuum, ac inter$e
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diff changeset
2506 <MARG><I>Tab.</I> 7. <I>fig.</I> 8.</MARG>
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diff changeset
2507 homogene&aelig; $int BAC, DEF, qu&aelig; habeant altitudines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2508 AC, EF &aelig;quales. Dico, imagines acceleratorum motuum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2509 e$$e inter $e, vt tempora dictorum $implicium motuum, vel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2510 reciproc&egrave; vt amplitudines ip$arum gene$um. Concipian-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2511 <pb n=72>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2512 tur imagines velocitatum $implici&utilde; motuum, $cilicet GHI
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2513 iuxta gene$im BAC, et MKL iuxta alter&atilde; gene$im DEF, &amp;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2514 quia, vtpot&egrave; homogene&ecedil;, $unt inter $e vt $patia &ecedil;qualia AC
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parents:
diff changeset
2515 ad EF, er&utilde;t ip$&aelig; imagines &ecedil;quales inter $e, c&utilde; ver&ograve; ob $imili
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2516 tudin&etilde; motuum e&aelig; ip$&aelig; imagines nectantur ex rationibus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2517 GI ad ML, &amp; ex ea, quam habet HI ad KL, $equitur e$$e
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2518 GI ad ML, vt KL ad IH, &amp; demum quia acceleratorum
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diff changeset
2519 motuum $patia &agrave; $implicibus imaginibus GHI, MKL ne-
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parents:
diff changeset
2520 ctuntur ex duplicata temporum HI ad KL, &amp; ex ea an pli-
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diff changeset
2521 <MARG>27. <I>huius.</I></MARG>
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parents:
diff changeset
2522 tudinum GI ad ML, $iue ex ea, quam habet KL ad HI, re-
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parents:
diff changeset
2523 linquitur, $patia acceleratis illis motibus confecta e$<*> i&ntail;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2524 $ola, vnicaq; ratione temporum HI ad KL, vel in ei &ecedil;qua-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2525 li ratione, reciproca amplitudinum imaginum ML ad GI,
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diff changeset
2526 vel gene$um DF ad BC. Quod &amp;c,
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diff changeset
2527 <C>PROP. XXX. THEOR. XXIV.</C>
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parents:
diff changeset
2528 <p>QV&aelig;cunque fuerint gene$es $imilium, $impliciumque
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parents:
diff changeset
2529 motuum, dum inter$e homogene&aelig;, $patia accelera-
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casties
parents:
diff changeset
2530 tis motibus ex illis $implicibus exacta nectentur
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parents:
diff changeset
2531 ex duplicata ratione altitudinum, &amp; reciproca amplitudi-
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parents:
diff changeset
2532 num earundem $implicium gene$um,
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parents:
diff changeset
2533 <p>Sint qu&aelig;cunque $imilium motuum gene$es BAC, KFG.
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diff changeset
2534 <MARG><I>Tab.</I> 7. <I>fig.</I> 6.</MARG>
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parents:
diff changeset
2535 Dico, $patia acceleratorum motuum, ab ijs $implicibus de-
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parents:
diff changeset
2536 riuantium, componi ex duplicata ratione altitudinum AC
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parents:
diff changeset
2537 ad FG, &amp; ex ratione extremarum velocitatum, $eu ampli-
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casties
parents:
diff changeset
2538 tudinum reciproc&egrave; $umptarum ip$arum gene$um: e$to alia
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casties
parents:
diff changeset
2539 gene$is DFG illis homogenea, &amp; motu pariter $imilis cum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2540 ij$dem gene$ibus. Eadem $it amplitudine &aelig;qualis BAC,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2541 &amp; altitudo eius $it FG, $patia acceleratorum motuum ex
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diff changeset
2542 <MARG>28. <I>huius.</I></MARG>
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diff changeset
2543 $implicibus gene$ibus &aelig;quales amplitudines haben<*>bus,
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diff changeset
2544 &amp; $imilium motuum BAC, DFG $unt in duplicata 'ratione
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parents:
diff changeset
2545 rectarum, $eu altitudinum AC ad FG, &amp; $patia accelera-
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diff changeset
2546 <MARG>29. <I>huius.</I></MARG>
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diff changeset
2547 <pb n=73>
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parents:
diff changeset
2548 torum motuum ex $implicibus gene$ibus, qu&aelig; $int in ea-
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parents:
diff changeset
2549 dem altitudine DFG, KFG, $unt in reciproca ratione am-
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diff changeset
2550 plitudinum, $eu primarum vel ocitatum KG ad DG, vel
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parents:
diff changeset
2551 BC; ex &aelig;quali igitur $patia acceleratorum motuum ex
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parents:
diff changeset
2552 propo$itis $implicibus gene$ibus BAC, KFG nectentur ex
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parents:
diff changeset
2553 ratione duplicata altitudinum AC ad FG, &amp; reciproca
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diff changeset
2554 amplitudinum KG ad BC earundem gene$um BAC,
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parents:
diff changeset
2555 KFG. Quod &amp;c.
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parents:
diff changeset
2556 <C><I>Scholium.</I></C>
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parents:
diff changeset
2557 <p><I>At quia in $patijs, qu&aelig; accelerato motu peraguntur; non
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diff changeset
2558 $eruatur ratio altitudinum gene$um $implicium, ex quo ori-
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diff changeset
2559 tur in hac methodo qu&aelig;dam percipiendi dificultas; ideo $e-
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diff changeset
2560 quenti problemate, alij$que iam notis veritatibus, rem plan&egrave;
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diff changeset
2561 illu$trabimus, ac $imul doctrina v$um trademus.</I>
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diff changeset
2562 <C>PROP. XXXI. PROB. VI.</C>
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diff changeset
2563 <p>EX datis $patijs accelerato motu confectis, cogniti$-
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2564 que primis, aut po$tremis $imilium, $impliciumque
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diff changeset
2565 motuum velocitatibus, reperire tempora ip$orum de-
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diff changeset
2566 cur$uum.
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diff changeset
2567 <p>Spatia motibus acceleratis exacta $unt C, D, &amp; velo-
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2568 <MARG><I>Tab.</I> 8. <I>Fig.</I> 1.</MARG>
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diff changeset
2569 tates, $eu amplitudines gene$um ponantur e$$e A, B, $cili-
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parents:
diff changeset
2570 cet A principio motus per C, &amp; B initio motus per D, qu&aelig;-
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parents:
diff changeset
2571 ritur ratio temporum, quibus exiguntur propo$ita $patia.
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diff changeset
2572 Vt A ad B, ita fiat C ad E, &amp; inter E, et D $umatur F me-
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diff changeset
2573 dia proportionalis. Dico ip$a tempora e$$e vt E ad F.
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diff changeset
2574 Componuntur $patia acceleratis motibus exacta ex ratio-
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diff changeset
2575 <MARG>27. <I>huiuij</I></MARG>
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diff changeset
2576 ne quadratorum temporum, &amp; ex ea amplitudinum, $eu
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diff changeset
2577 homologarum velocitatum in $implicibus motibus, $imili-
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diff changeset
2578 <MARG><I>lem. pr.</I> 3. <I>pri-
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diff changeset
2579 mi huius.</I></MARG>
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diff changeset
2580 bu$que $umptarum; &amp; ideo temporum quadrata necten-
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diff changeset
2581 tur ex ratione $patiorum C ad D, &amp; ex reciproca ampli-
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diff changeset
2582 <pb n=74>
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diff changeset
2583 tudinum E ad C; temporum igitur quadrata erunt vt E ad
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diff changeset
2584 D, ip$a ver&ograve; tempora vt E ad F. Quod &amp;c.
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diff changeset
2585 <C>PROP. XXXII. PROB. VII.</C>
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2586 <p>EXdatis $patijs accelerato motu tran$actis, datis item
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parents:
diff changeset
2587 primis velocitatibus $imilium, $impliciumque mo-
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diff changeset
2588 tuum, inuenire altitudines $implicium gene$um, ex quibus
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2589 <MARG><I>Tab.</I> 7. <I>fig.</I> 1.
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diff changeset
2590 30. <I>huius.</I></MARG>
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2591 propo$ita $patia effecta $unt.
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diff changeset
2592 <p>Spatia $int E, D reliquis, vt $upra, manentibus: quoniam
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diff changeset
2593 $patia accelerato motu tran$acta componuntur ex ratio-
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diff changeset
2594 nibus amplitudinum gene$um $implicium, $imiliumqu&etail;
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diff changeset
2595 motuum reciproc&egrave; $umptarum B ad A, $iue E ad C, &amp; ex
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diff changeset
2596 ea quadratorum altitudinum ip$arum gene$um; erit ratio
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diff changeset
2597 dictarum altitudinum duplicata C ad D; quare F, $i $it me-
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diff changeset
2598 dia proportionalis, non inter E, &amp; D (vt antea po$uimus)
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diff changeset
2599 $ed inter C ad D; erit $an&egrave; C ad F ratio altitudinum gene-
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diff changeset
2600 $um $implicium, $imiliumque motuum, quam quereba-
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diff changeset
2601 mus.
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diff changeset
2602 <C><I>Exemplum primum.</I></C>
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2603 <p>SI idem graue naturaliter cadens percurrerit &agrave; quiete
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diff changeset
2604 duo $patia; tempora erunt in ratione $ubduplicat&atail;
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parents:
diff changeset
2605 corundem $patiorum.
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2606 <p>Ex Cor. pr: 4. huius con$tat rectangula e$$e gene$es $im-
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diff changeset
2607 plicium motuum grauium naturaliter de$cendentium, &amp;
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2608 ex def. 7. primi liquet ea$dem gene$es e$$e motuum $imi-
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diff changeset
2609 lium. Cumque eiu$dem mobilis naturaliter cadentis ve-
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parents:
diff changeset
2610 locitas &agrave; quiete $it vna, eademque; $implices motus erunt
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diff changeset
2611 ij, vt gene$um $imilium, $impliciumque motuum amplitu-
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diff changeset
2612 dines &aelig;quales $int, proptereaque, vt in figura pr&aelig;cedentis
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diff changeset
2613 propo$itionis &aelig;quales erunt C, E, atque adeo $patiu&mtail;
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parents:
diff changeset
2614 C, $iue E ad D erit in duplicata ratione temporum E ad F.
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2615 <pb n=75>
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2616 <C><I>Exemplum II.</I></C>
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2617 <C>PROP. XXXIV. THEOR. XXVII.</C>
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2618 <p>TEmpora $imilium vibrationum $unt in $ubduplicata
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parents:
diff changeset
2619 ratione arcuum exactorum, $eu longitudinum pen-
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diff changeset
2620 dulorum, quorum $unt vibrationes. Sint grauia pendula
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parents:
diff changeset
2621 LA, LF, qu&aelig; ab eadem recta LF di$cedentia currant $u$-
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2622 <MARG><I>Tab.</I> 8. <I>fig.</I> 2.</MARG>
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diff changeset
2623 pen$a ex L duos $imiles arcus circulares FI, AC. Dico
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diff changeset
2624 tempora horum de$cen$uum e$$e in ratione $ubduplicat&atail;
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diff changeset
2625 arcuum FI, AC, $eu longitudinum filorum, aut ha$tularum
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diff changeset
2626 FA, LA. Ducamus quamcumque rectam LBG, erit AB
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2627 ad BC, vt FG ad GI, &amp; cum pr&aelig;terea velocitates pendu-
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diff changeset
2628 lorum a quiete in A, F $int &aelig;quales, pariterque velocita-
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diff changeset
2629 tes &aelig;quales a quiete in B, G; erit velocitas in A ad veloci-
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diff changeset
2630 tatem in B, vt velocitas in F ad velocitatem in G, quare
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2631 con$ideratis arcubus ABC, FGI, vt altitudines rect&ecedil;, (qu&aelig;
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2632 item forent in B, G proportionaliter $ect&ecedil;) gene$um $imi-
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2633 <MARG><I>Def.</I> 7. <I>pri<*></I></MARG>
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diff changeset
2634 lium $impliciumque motuum, quarum amplitudines &aelig;qua
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diff changeset
2635 les $unt, erunt $patia in acceleratis decur$ubus per FI, AC
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diff changeset
2636 in ratione duplicata temporum, $cilicet ip$i arcus, aut lon-
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diff changeset
2637 gitudines LF, LA erunt in ratione duplicata temporu&mtail;.
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diff changeset
2638 Quod &amp;c.
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diff changeset
2639 <p>Idem demon$tratum e$$et beneficio imaginum, qu&aelig; vt-
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diff changeset
2640 pote eorundem illorum motuum $implicium, forent etiam
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diff changeset
2641 $imilium, &amp; $unt amplitudines &aelig;quales, etenim e&aelig;de&mtail;
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diff changeset
2642 $unt, ac gene$um ergo rur$us $patia, hoc e$t arcus ABC,
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diff changeset
2643 FGI, nempe longitudines filorum IF, AC erunt in ratione
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diff changeset
2644 duplicata temporum. Quod &amp;c.
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diff changeset
2645 <pb n=76>
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diff changeset
2646 <C><I>Scholium.</I></C>
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2647 <p><I>Vides, qu&agrave;m breuiter rei di$ficillim&aelig; demon$tr ationem at-
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diff changeset
2648 tulimus, nec dubium, quin illa extendi queat ad qua$cum-
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diff changeset
2649 que lineas decur$uum, dummodo $imiles, ac $imiliter po$itas in
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diff changeset
2650 ij $dem, vel &aelig;qualibus ab horizonte planis elenatis, quemad-
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diff changeset
2651 modum Dominus Viuianus pulcherrim&egrave; propo$uit.</I>
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diff changeset
2652 <C><I>Exemplum III.</I></C>
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diff changeset
2653 <C>PROP. XXXV. THEOR. XXVIII.</C>
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diff changeset
2654 <p>TEmpora lationum &agrave; quiete per plana eandem eleua-
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diff changeset
2655 <MARG><I>Tab.</I> 8. <I>fig.</I> 3.</MARG>
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diff changeset
2656 tionem habentia $unt homolog&egrave; vt longitudines
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diff changeset
2657 planorum.
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diff changeset
2658 <p>Sint plana AB, AC eandem eleuationem AD habentia.
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diff changeset
2659 Dico tempus lationis per AC ad id per AB e$$e vt AC ad
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diff changeset
2660 AB. (h&aelig;c Torricellij propo$itio, expo$itioq; e$t, hancque
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diff changeset
2661 eandem veritatem ex no$tris principijs demon$trare vis&utilde;
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diff changeset
2662 e$t, non vt de re illa dubitemus, imm&ograve; contr&agrave;, qu&ograve;d de e&atail;
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diff changeset
2663 plen&egrave; $atisfacti $imus, ex eo rur$us demon$trandam $u$ce-
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diff changeset
2664 pimus, vt exinde methodus no$tra, qu&agrave;m vera $it, eluce$-
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diff changeset
2665 cat) Momentum de$cen$us inplano AC ad id de$cen$us $u-
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diff changeset
2666 <MARG><I>Tor. pr.</I> 2. <I>de
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diff changeset
2667 motu graui&utilde;.</I></MARG>
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diff changeset
2668 per plano AB e$t vt AB ad AC; $unt autem de$cendenti&utilde;
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diff changeset
2669 grauium, etiam $uper planis inclinatis motus, quos $impli-
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diff changeset
2670 ces appellamus, inter $e $imiles, nempe quorum gene$es
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diff changeset
2671 <MARG><I>Cor pr.</I> 4.
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diff changeset
2672 <I>huius.</I></MARG>
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diff changeset
2673 $unt rectangula; ergo habebimus $implices gene$es, vnam,
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diff changeset
2674 cuius altitudo AC amplitudoque AB; alteram, cuius am-
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diff changeset
2675 plitudo AC, altitudo autem AB; itaque propo$itis $patijs
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diff changeset
2676 AC, AB, primi$que velocitatibus AB, AC, $i fiat AB ad AC
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diff changeset
2677 vt CA ad EA, erit EA ad AB duplicata t&etilde;porum, &amp; ideo
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diff changeset
2678 <MARG>31. <I>i<*></I> 27. <I>hu.</I></MARG>
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diff changeset
2679 ratio temporum per AC, AB erit CA ad AB. Quod &amp;c.
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diff changeset
2680 <pb n=77>
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diff changeset
2681 <C><I>Exemplum IV.</I></C>
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diff changeset
2682 <C>PROP. XXXVI. THEOR. XXIX.</C>
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diff changeset
2683 <p>II$dem pror$us manentibus demon$trarunt Gallileus, ac
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diff changeset
2684 Torricellius, gradus velocitatum acqui$itos in B, et C
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diff changeset
2685 eiu$dem mobilis de$cendentis &agrave; quiete in A pares e$$e;
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diff changeset
2686 idip$um nos o$tendemus.
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diff changeset
2687 <p>Cum tempora $int vt AC ad AB, &amp; velocitates &agrave; quie-
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diff changeset
2688 te in ratione reciproca temporum, $cilicet vt AB ad AC,
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2689 <MARG><I>Tab.</I> 8. <I>fig.</I> 4.
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diff changeset
2690 33. <I>huius.</I></MARG>
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diff changeset
2691 $int deinde velocitates e&aelig; vt amplitudines imaginum $im-
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diff changeset
2692 plicium, $imiliumque illorum motuum (nam amplitudines
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diff changeset
2693 imaginum velocitatum $unt pror$us e&aelig;dem, ac ill&aelig; gene-
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diff changeset
2694 $um) erunt ip$&aelig; imagines $implicium motuum &aelig;quales;
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diff changeset
2695 nam tempora, qu&aelig; $ummuntur vt altitudines imaginum
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diff changeset
2696 reciprocantur, vt dictum e$t, amplitudinibus, $eu primis &agrave;
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diff changeset
2697 <MARG>4. <I>huius.</I></MARG>
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diff changeset
2698 quiete velocit atibus, at in motibus acceleratis ip$&aelig; inte-
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diff changeset
2699 gr&aelig; imagines $implicium motuum $unt loco graduum ve-
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diff changeset
2700 locitatum in extremo $patiorum acqui$itorum; ergo in B, et
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diff changeset
2701 C gradus velocitatum &aelig;quales erunt.
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2702 <C>PROP. XXXVII. THEOR. XXX.</C>
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2703 <p>SI &aelig;qualia pondera, $u$pen$a $int ex filis, quorum par-
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diff changeset
2704 tes inter$e &aelig;quales, pr&aelig; tractione &aelig;qualiter elongen-
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diff changeset
2705 ter tempora in reditu ip$orum filorum, cum ab ip$is graui-
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diff changeset
2706 bus $tatim liberantur, &aelig;qualia erunt. Hoc prim&ugrave;m dem&otilde;-
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2707 <MARG><I>Tab.</I> 8. <I>fig.</I> 5.</MARG>
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2708 $trabimus alia via, tum methodo no$tra, vt de ea aliud
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diff changeset
2709 exemplum tradamus. Sint funiculi AB, DC, &amp; ex ijs
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diff changeset
2710 pendeant &aelig;qualia grauia B, C, adeo vt $umptis hinc ind&egrave;
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diff changeset
2711 partibus &aelig;qualibus eorundem $uniculorum, con$tet ip$as
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diff changeset
2712 &aelig;qualiter ab ip$is grauibus trahi, atque produci. Dico, $i
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2713 elongationes $int HB, GC, &amp; omnibus $ic $tantibus pon-
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2714 <pb n=78>
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parents:
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2715 dera $ubmoueantur ex B, et C funiculis e&aelig;$is, fore vt e&aelig;-
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2716 dem extremitates re$tituantur in H, et G &aelig;qualibus tem-
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2717 poribus. Sit AE &aelig;qualis DC, erit porr&ograve; elongatio facta
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parents:
diff changeset
2718 per idem graue B, qu&aelig; $it EF, &aelig;qualis GC; propterea li-
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parents:
diff changeset
2719 beratis funiculis ad B, et C, eodem tempore re$tituetur C
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2720 in G, ac E in F, quo tempore etiam B in H re$titutum fue-
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2721 rit; nam vno puncto in primum $uum locum redito, etiam
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parents:
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2722 alia $ingula in $uum locum perueni$$e, opportebit.
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diff changeset
2723 <C><I>Exemplum.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2724 <p>HAc occa$ione de funiculis erit non iniucunda di$er-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2725 tatio, remque $ic adhuc intactam promouebimus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2726 $imulque demon$trabimus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2727 <p>Idip$um propo$itum no$tris principijs $ic demon$tra-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2728 mus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2729 <p>Sint ead&etilde;, qu&aelig; $upra, $cilicet conceptis in filo AB quot-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2730 libet partibus inter$e &aelig;qualibus, l&otilde;gitudin&etilde;que totam im-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2731 plentibus, h&aelig; $ingul&aelig; &aelig;qualiter &agrave; pondere B trahentur,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2732 eritque BH $umma omnium dictarum partium elongatio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2733 num, &amp; eodem pacto EF erit $umma elongationum parti&utilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2734 omnium in AE contentarum, ab eodemque pondere effe-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2735 ctarum; propterea vt AB ad BH, ita erit AE ad EF; quamo
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2736 brem velocitas etiam puncti B $ublato pondere B erit ad
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2737 velocit atem puncti E ob eandem detractionem, vt BH ad
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2738 EF, vel BA ad EA (nam quot $unt partes concept&ecedil; i&ntail;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2739 vtraque fili longitudine, totidem $unt etiam impetus inter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2740 $e &aelig;quales) idem o$tenderemus $i loco ponderis B, minus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2741 quodcumque $u$penderemus, vt $cilicet puncta B, et E ad
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2742 quemuis locum $uperius remanerent, librarenturque cum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2743 re$i$tentijs parti&utilde; e&ograve; elongatarum, ergo tran$itus ex B in H,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2744 <MARG><I>pr.</I> 4. <I>huius.</I></MARG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2745 &amp; puncti E in F $ubducto pondere B erunt motus $imilium
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2746 $impliciumque; $ed motus ex C in G exempto pondere C
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2747 e$t pror$us idem, ac motus E in F, ergo motus $imiles, ac
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2748 <pb n=79>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2749 $implices ex B in H, &amp; ex C in G, ex quibus fiunt accele-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2750 rati, gene$es habebunt, quarum prim&aelig; velocitates, $eu am-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2751 plitudines proportionales $unt altitudinibus earundem,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2752 $patijs nimirum CG, BH accelerato motu exigendis; qua-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2753 mobrem componentur ex ratione ip$arum velocitatum,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2754 $eu amplitudinum CG ad BH, &amp; ex ea quadratorum tem-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2755 porum, qu&aelig; proinde &aelig;qualitatis erit; itaque etiam huius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2756 $ubduplicata; hoc e$t tempora in tran$itibus accelarato
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2757 motu exactis, erunt paria.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2758 <C><I>Corollarium.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2759 <p><I>Hinc patet, vbi &aelig;qu&egrave; cra$$is filis eiu$demque materiei vel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2760 cedenti&aelig; $u$pen$a $int &aelig;qualia pondera, tunc primas velocita-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2761 tes, $ubductis ponderibus, fore in eadem ratione elongation&utilde;,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2762 vel longitudinum filorum.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2763 <C>PROP. XXXVIII. THEOR. XXXI.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2764 <p>SI extremitatibus funiculorum ex vna parte firmator&utilde;,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2765 ac eandem cra$$itiem habentium, nec non eiu$dem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2766 c&aelig;denti&aelig; exi$tentium, fuerint $u$pen$a &aelig;qualia pondera,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2767 qu&aelig; inde ij$dem longitudinibus $eruatis, quomodo opor-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2768 tet tollantur, erunt $patia recur$uum, temporibus $impli-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2769 cium motuum exacta in ratione longitudinum pendulo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2770 rum.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2771 <p>Sit funiculus AC &aelig;qu&egrave; cra$$us ac BD, &amp; $u$pen$is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2772 hinc inde ponderibus &aelig;qualibus, elongatio primi funiculi
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2773 $it CE, &amp; alterius $it DF. Dico $patia temporibus $impli-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2774 cium imaginum, ab extremitatibus $olutis exacta, fore i&ntail;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2775 ratione longitudinum ip$orum funiculorum.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2776 <p>Iam con$tat CE ad DF e$$e, vt AC ad BD, in qua ratione
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2777 $unt etiam velocitates &agrave; quiete, dum pondera $ubduceren-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2778 tur ex E, et F, vel ex alijs punctis quibu$cunque $i &aelig;qualia
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2779 <pb n=80>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2780 pondera $u$pen$a fui$$ent maioris, vel minoris ponderis,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2781 $ic enim concipiuntur gene$es $imilium, $impliciumque
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2782 motuum, quarum altitudines &aelig;quantur elongationibus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2783 funiculorum; propterea $patia recur$uum temporibus $im-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2784 plicium motuum exacta, nectentur ex rationibus duplicata
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2785 CE ad DF, hoc e$t AC ad BD, &amp; ex reciproca filorum,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2786 $cilicet BD ad AC, qu&aelig; ratio, vti diximus, e$t reciproc&atail;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2787 primarum velocitatum, $eu amplitudinum gene$um $impli-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2788 cium, ergo ip$a $patia in reditu filorum ab extremitatibus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2789 $olutis exacta, erunt vt AC ad BF, $eu vt CE ad DF.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2790 Quod &amp;c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2791 <C>PROP. XXXIX. THEOR. XXXI.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2792 <p>TEmpora $implicium, $imiliumque dictorum motuum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2793 $unt &aelig;qualia.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2794 <p>Nam cor. 2. pr. 8. huius primi demon$tratum e$t, tem-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2795 pora $implicium, $imiliumque motuum componi ex ratio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2796 ne $patiorum, $eu altitudinum gene$um, &amp; reciproca pri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2797 marum, aut extremarum velocitatum, $eu amplitudinum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2798 gene$um: $unt autem altitudines gene$um tractiones, $eu
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2799 elongationes funiculorum, qu&aelig; $unt vt longitudines funi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2800 culorum, ergo tempora &aelig;qualia erunt.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2801 <C><I>Corollarium.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2802 <p><I>Con$tat, tempora a $implicium gene$um in tractionibus fu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2803 niculorum, e$$e compo$ita ex ratione elongationum funiculo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2804 rum, &amp; ex reciproca primarum velocitatum.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2805 <pb n=81>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2806 <C><I>Scholium.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2807 <p><I>Superioris propo$itionis veritas concordat cum prop.</I> 37. <I>bu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2808 ius, in eo tant&ugrave;m variatur, quod ibi ponuntur data $pati&atail;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2809 elongitiones funiculorum, hic ver&ograve; tempora $impliciu&mtail;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2810 motuum, &amp; quia elungationes o$ten$&aelig; $unt proportionales $pa
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2811 tijs nunc exactis, manife$tum e$t, no$tri iuris e$$e mod&ograve; $patia
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2812 acceleratis motibus exact a ex temporibus $implicium motu&utilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2813 datis concludere, mod&ograve; contr&agrave;, ex $patijs altitudinibus gene-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2814 $um proportionalibus, qua item data $unt, tempora inuenire,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2815 qua proinde methodus mihi videtur ampli$$ima.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2816 <C>PROP. XXXX. THEOR. XXXIII.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2817 <p>SI eiu$dem cra$$itiei funiculis pondera dependeant, qu&ecedil;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2818 $int in ratione reciproca longitudinum ip$orum funi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2819 culorum, $patia temporibus gene$um $implicium motuum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2820 exacta erunt in ratione duplicata elongationum.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2821 <MARG><I>Tab.</I> 8. <I>Fig.</I> 6.</MARG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2822 <p>N&atilde; $i $it p&otilde;dus E ad F $icuti l&otilde;gitudo DB ad CA, &amp; $int,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2823 cra$sities funiculor&utilde; &aelig;quales erit $an&egrave; ratio, qu&aelig; c&otilde;poni-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2824 tur ex ratione funiculor&utilde;, &amp; ex ea p&otilde;derum, &aelig;qualitatis; ob
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2825 idque gene$es $implici&utilde; motu&utilde;, quar&utilde; altitudines CE, DF
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2826 habeb&utilde;t amplitudines, n&etilde;pe primas velocitates inter$e&ecedil;qua
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2827 les (nam cum pondera erant &aelig;qualia, prim&aelig; velocitates
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2828 proportionabantur longitudinibus funiculor&utilde;, ideo, cum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2829 <MARG>28. <I>huius.</I></MARG>
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parents:
diff changeset
2830 pondera reciprocantur longitudinibus ij$dem, $eu viribus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2831 funiculorum, fit vt prim&aelig; velocitates &aelig;quales reddantur)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2832 cum ergo ita $it, $patia recur$uum temporibus imaginu&mtail;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2833 $implicium &amp; accelerato motu confecta erunt in ratione
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2834 duplicata elongationum.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2835 <pb n=82>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2836 <C><I>Corollarium.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2837 <p><I>Cum ex eadem pr.</I> 28. <I>huius, eadem $patia $int vt quadra-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2838 ta temporum, erunt ip$a tempera in ratione $ubduplicat&atail;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2839 elongationum.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2840 <C>PROP. XXXXI THEOR. XXXIV.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2841 <p>SI funiculis &aelig;qualem cra$$itiem habentibus fuerint $u$-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2842 <MARG><I>Tab.</I> 8. <I>Fig.</I> 6.</MARG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2843 pen$a in&aelig;qualia pondera, $patia, qu&aelig; acceleratis mo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2844 tibus, ac temporibus gene$um $implicium recurruntur ne-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2845 ctentur ex ratione duplicata elongationum, &amp; ex duabus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2846 reciproc&egrave; $umptis rationibus, nempe longitudinum prima-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2847 rum funiculorum, antequam pondera $u$penderentur; &amp;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2848 ip$orum ponderum.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2849 <p>In antecedenti figura illud primum $atis patet, qu&ograve;d $i
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2850 loco ponderis F $u$pen$um fui$$et pondus aliud grauius,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2851 aut leuius, prior velocitas in a$cen$u fili, $eu funiculi, aut
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2852 chord&aelig; aucta, vel imminuta fui$$et pro magnitudine pon-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2853 deris $ub$tituti; quamobrem priores velocitates ex in&aelig;qua
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2854 litate ponderum eidem chord&aelig; $u$pen$orum dependentes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2855 forent, vt ip$a pondera; ver&ugrave;m cum $uppo$itis funiculis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2856 &aelig;qualia pondera $u$pen$a veniunt, prim&aelig; velocitates $unt
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2857 <MARG><I>Cor. pr.</I> 37.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2858 <I>huius.</I></MARG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2859 vt longitudines funiculorum, ergo velocitates prim&aelig;, cum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2860 in&aelig;qualia $unt pondera, qu&aelig; $ubtrahuntur, nectentur ex
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2861 ratione longitudinum funiculorum, &amp; ex ea ponderum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2862 in&aelig;qualium: qu&aelig;cumque igitur $it tractio DF, gene$es ha-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2863 bebimus $imilium $impliciumque motuum, vnam, cuius al-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2864 titudo CE, &amp; alteram habentem altitudinem DF, &amp; $unt
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2865 earundem gene$um amplitudines, $eu prim&aelig; velocitates
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2866 in ratione compo$ita funiculorum AC ad BD, &amp; ponderis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2867 <MARG>30. <I>huius.</I></MARG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2868 pendentis ex E ad pondus $u$pen$um in F; ergo $patia ac-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2869 celeratis motibus tran$acta temporibus gene$um $implici&utilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2870 <pb n=83>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2871 nectentur ex ratione dublicata elongationum, $iue altitu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2872 dinum gene$um, &amp; ex duabus rationibus reciproc&egrave; $um-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2873 ptis funiculorum AC ad BD, &amp; ponderum E ad F.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2874 Quod &amp;c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2875 <C>PROP. XXXXII. THEOR. XXXV.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2876 <p>II$dem po$itis, $i $patia recur$uum erunt ip$&aelig; elongatio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2877 nes, tempora, quibus ab extremitatibus $olutis recur-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2878 runtur, erunt in ratione $ubduplicata eorundem. Nam cum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2879 gene$es $imilium, $impliciumque motuum $int &aelig;qu&egrave; am-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2880 pl&aelig;, erunt, tempora in ratione $ubduplicata imaginum,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2881 $eu $patiorum acceleratorum motuum, $unt ver&ograve; $patia
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2882 ip$&aelig; elongationes; ergo &amp;c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2883 <C>PROP. XXXXIII. THEOR. XXXVI.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2884 <p>CHord&aelig; non eiu$dem cra$$itiei, eiu$dem tamen mate-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2885 ri&aelig;, ac longitudinis, tunc &aelig;qu&egrave; trahentur <*>bi $u$p&etilde;-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2886 $a pondera cra$$itut inibus proportionalia fuerint. Nam
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2887 cra$$ior chorda pote$t concipi compo$ita ex funiculis eiu$
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2888 dem cra$$itiei alterius chord&aelig;, $i illa huius fuerit multiplex,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2889 &amp; $i partes exilior funiculus fuerit alterius cra$$ioris, erit
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2890 cra$$ities alicuius alterius funiculi, qu&aelig; pluries accept&atail;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2891 con$tituere poterit vtranque cra$$itiem funiculorum pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2892 po$itorum (h&igrave;c enim non accidit enumerare cra$$ities in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2893 ter$e irrationales, quippe quia, quod de iam dictis o$ten-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2894 derimus, de his quoque facil&egrave; e$t iudicare, $ec&ugrave;s e$$emus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2895 longi, quam par e$t, poti$$im&ugrave;m cum h&aelig;c pr&aelig;ter in$titut&utilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2896 adijciantur, &amp; quidem vt con$tet, quomodo methodus i$ta
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2897 no$tra facilis $it, ac vtili$$ima) quapropter $i cuique acce-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2898 ptarum &aelig;qualium chordarum, pondera &aelig;qualia $u$pen$a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2899 $int, porr&ograve; h&aelig;c omnes &aelig;qu&egrave; trahentur ab ip$is &aelig;qualibus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2900 ponderibus, &amp; $ic etiam compo$ita, nempe choid&aelig; pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2901 <pb n=84>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2902 po$it&aelig;; $untque ita pondera in eadem ratione cra$$itierum,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2903 $icut propo$uimus; ergo patet propo$itum.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2904 <C>PROP. XXXXIV. THEOR. XXXVII.</C>
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diff changeset
2905 <p>SI fuerint eiu$dem materi&aelig; funiculi, &amp; $int illis $u$pen$a
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diff changeset
2906 pondera cra$$itiebus proportionalia, ratio $patiorum
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diff changeset
2907 in reditibus accelerato motu exactorum, t&etilde;poribus $im-
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diff changeset
2908 <MARG><I>Tab.</I> 8. <I>fig.</I> 6.
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diff changeset
2909 42. <I>huius.</I></MARG>
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diff changeset
2910 plicium gene$um, erit eadem ac funiculorum.
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diff changeset
2911 <p>Nam, vt in pr&aelig;cedenti figura, erit tractio CE ad DF ita
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diff changeset
2912 <MARG><I>Cor. pr.</I> 37.
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diff changeset
2913 <I>huius.</I></MARG>
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diff changeset
2914 AC ad BD, vel AE ad BF, $unt autem prim&aelig; velocitates,
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diff changeset
2915 $eu amplitudines gene$um $implicium, $imiliumque motu&utilde;
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diff changeset
2916 in ratione funiculorum, ergo decur$uum $patia motibus
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diff changeset
2917 <MARG>27. <I>huius.</I></MARG>
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parents:
diff changeset
2918 acceleratis exacta nectentur ex ratione duplicata altitu-
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diff changeset
2919 dinum gene$um $implicium, nempe duplicata funiculor&utilde;,
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parents:
diff changeset
2920 &amp; reciproca amplitudinum, $untque ip$&aelig; amplitudines
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parents:
diff changeset
2921 homolog&egrave; vt longitudines funiculorum, ergo relinquitur
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diff changeset
2922 vt ip$a $patia $int in vnica ratione longitudinum funicu-
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parents:
diff changeset
2923 lorum.
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parents:
diff changeset
2924 <p>Qu&ograve;d $i $patia recur$uum ponantur ip$&aelig; tractiones, vel
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parents:
diff changeset
2925 longitudines funiculorum, o$tendetur tempora e$$e &aelig;qua-
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parents:
diff changeset
2926 lia, quemadmodum &aelig;qualia $unt tempora $uperius pro-
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parents:
diff changeset
2927 po$ita $implicium gene$um.
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parents:
diff changeset
2928 <C>PROP. XXXXV. THEOR. XXXVIII.</C>
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diff changeset
2929 <MARG><I>Tab.</I> 8. <I>fig.</I> 7.</MARG>
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diff changeset
2930 <p>SI eiu$dem materiei quibu$cunque funiculis aligentur
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parents:
diff changeset
2931 qu&aelig;cunque pondera, ijs $ublatis a$cen$uum $patia ab
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diff changeset
2932 extremitatibus $olutis exacta temporibus gene$um $impli-
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parents:
diff changeset
2933 cium, ijs nempe qu&aelig; impenderentur in motibus iuxta $im-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2934 plices gene$es, erunt in ratione compo$ita quadratorum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2935 elongationum chordarum, ex ea cra$$itierum, &amp; ex duabus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2936 reciproc&egrave; $umptis rationibus, nempe longitudinum fu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2937 niculorum antequam traherentur; &amp; $u$pen$orum ponde-
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diff changeset
2938 rum.
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parents:
diff changeset
2939 <pb n=85>
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parents:
diff changeset
2940 <p>Funiculi AB, GH trahantur &agrave; ponderibus quibu$cunque
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parents:
diff changeset
2941 C, I in C, et I. Dico $i exempta $int pondera, fore, vt $patia
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2942 qu&aelig; acceleratis motibus exiguntur ab extremitatibus $o-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2943 lutis C, I $int in ratione compo$ita ex duplicata IH ad BC,
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parents:
diff changeset
2944 cra$$itudinis ad cra$$itudinem funiculorum AB, GH; dein-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2945 de ex funiculi longitudine HG ad longitudinem AB, pon-
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parents:
diff changeset
2946 deri$que I ad pondus C. Intelligatur funiculus, $eu chor-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2947 da, &aelig;que cra$$a, ac $imiliter cedens, qu&agrave;m GH (id quod
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2948 $emper intelligimus quoties funiculi, inter$e comparantur)
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parents:
diff changeset
2949 $ed &aelig;qu&egrave; longa, ac AB, $itque illi pondus F adiectum, ad
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2950 quod C eandem habeat rationem, ac cra$$ities AB ad cra$-
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parents:
diff changeset
2951 $itiem DE, con$tat elongationem EF &aelig;qualem fieri ip$i
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parents:
diff changeset
2952 CB, &amp; cum prim&aelig; velocitates, $eu amplitudines &aelig;qu&egrave; al-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2953 tarum gene$um $imilium, $impliciumque motuum $int eti&atilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2954 &aelig;quales, $patia decur$uum acceleratis motibus exacta er&utilde;t
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2955 pror$us &aelig;qualia; $unt ver&ograve; funiculi DE, GH eiu$dem cra$-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2956 $itiei, ei$que $unt $u$pen$a duo'pondera in&aelig;qualia F, I; ergo
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2957 decur$uum $patia ab extremitatibus $olutis exacta nect&etilde;-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2958 tur ex ratione duplicata elongationum FE, $eu CB ad IH,
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parents:
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2959 ex ratione, quam habent longitudines funiculorum HG ad
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parents:
diff changeset
2960 DE, $eu AB, &amp; ex ea ponderum I ad F; ver&ugrave;m pondera I
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2961 ad F nectuntur ex rationibus ponderum I ad C et C ad F,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2962 qu&aelig; po$trema e$t ratio cra$$itiei funiculi AB ad craffitiem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2963 funiculi DE, $eu GH; ergo vt propo$uimus $patia accele-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2964 ratis motibus exacta, nectentur ex rationibus quadrator&utilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2965 CB ad HI; cra$$itudinum funiculorum AB, GH; ponder&utilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2966 I ad C, &amp; longitudinum HG ad AB. Quod &amp;c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2967 <C>PROP. XXXXVI. THEOR. XXXIX.</C>
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parents:
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2968 <p>TEmpora gene$um $implicium, dum chordis $u$pen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2969 duntur qu&aelig;cunque grauia, nectuntur, ex ratione
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2970 elongationum funiculorum, &amp; ex contrari&egrave; $umptis ratio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2971 nibus, cra$$itudinum, longitudinumque funiculorum, nec
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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2972 <pb n=86>
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parents:
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2973 non ponderum funiculis $u$pen$orum.
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parents:
diff changeset
2974 <p>Nam Cor: 2. pr. 8. primi demon$tratum e$t, tempor&atail;
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parents:
diff changeset
2975 $implicium $imiliumque motuum componi ex ratione $pa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2976 tiorum, $eu altitudinum gene$um, &amp; reciproca primarum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2977 velocitatum, $eu amplitudinum gene$um, $unt autem alti-
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parents:
diff changeset
2978 tudines gene$um tractiones, $eu elongationes funiculorum;
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parents:
diff changeset
2979 velocitatesver&ograve; prim&aelig; nectuntur ex rationibus cra$$itudi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2980 num, &amp; ex ea longitudinum funiculorum antequam tra-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2981 herentur (hoc enim $ubinde o$tendemus) ergo tempora
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2982 propo$ita $implicium gene$um, dum chordis allig&atilde;tur qu&ecedil;-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2983 cunque in&aelig;qualia pondera, componentur ex rationibus
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parents:
diff changeset
2984 elongationum funiculorum, &amp; ex contrari&egrave; $umptis cra$$i-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2985 tudinum, longitudinumque funiculorum, &amp; ponderum.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2986 <C><I>A&szlig;umptum.</I></C>
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parents:
diff changeset
2987 <MARG><I>Tab.</I> 8. <I>fig.</I> 7.</MARG>
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parents:
diff changeset
2988 <p>VEr&ugrave;m prim&aelig; velocitates in ij$dem chordis componi
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2989 ex ratione cra$$itudinum, longitudinum funiculor&utilde;,
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parents:
diff changeset
2990 &amp; $u$pen$orum ponderum, $ic o$tendemus,
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parents:
diff changeset
2991 <p>Quoniam in eadem po$trema figura velocitas, qua&mtail;
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parents:
diff changeset
2992 haberet funiculus AB ex liberatione ponderis e$t &aelig;qualis
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parents:
diff changeset
2993 velocitati, quam haberet alius funiculus, vbi hic etiam li-
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parents:
diff changeset
2994 <MARG>43. <I>huius.</I></MARG>
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2995 beraretur &agrave; pondere, $cilicet cum pondera cra$$itiebus fu-
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diff changeset
2996 niculorum proportionalia $unt, &amp; ip$i funiculi &aelig;qu&egrave; longi;
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diff changeset
2997 velocitas funiculi DE &agrave; pondere F ad velocitatem eiu$d&etilde;
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diff changeset
2998 <MARG><I>Ex</I> 41. <I>huius.</I></MARG>
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diff changeset
2999 funiculi, $i loco ponderis F $ub$titutum e$$et aliud &aelig;quale
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diff changeset
3000 ip$i I, e$$et vt pondus F ad $ub$titutum, $eu ad I, e$t autem
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diff changeset
3001 velocitas eiu$dem funiculi DE, dum fui$$et pondus ei $u$-
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diff changeset
3002 pen$um &aelig;quale I ad velocitatem funiculi GH a pondere I
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diff changeset
3003 vt longitudo DE ad GH; ergo patet propo$itum.
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diff changeset
3004 <pb n=87>
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diff changeset
3005 <C><I>Scholium.</I></C>
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diff changeset
3006 <p><I>Quod hucu$que ostendimus in funiculis ponderibus de-
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diff changeset
3007 grauatis, non ab$imili modo pr&aelig;st abimus in chordis ad vtr&atilde;-
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diff changeset
3008 que extremitatem firmatis, &amp; adductis, hoc tantum di$crimi-
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diff changeset
3009 ne, vt $i in ijs pondere $ublato, motus extremitatis $olut&aelig; at-
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diff changeset
3010 tendebatur, h&igrave;c media parte attract&acirc; chord&acirc;, &amp; $ubinde $ui
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diff changeset
3011 iuris relict&acirc;, vibrationem eius ob$eruamus, &amp; equidem illa
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3012 omnia in hunc finem o$tendimus, quippe ab hac re, plurima
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3013 vtili$$im&aelig;que veritates manere po$$unt. Nam de arcubus po$$es
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diff changeset
3014 pulcherrima in$titui ratio, &amp; qui vellet armonicorum $ono-
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diff changeset
3015 rum, vel vocum per chordarum vibrationes editarum, tempo-
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3016 ra, cum $oni ad aures perueniunt, inue$tigare, reor non aliam
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diff changeset
3017 viam, qu&agrave;m hanc ingredi nos debere, atque ind&egrave; con$onantia-
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diff changeset
3018 rum forta$$e naturam pertipere po$$e, vt primus <*>ui<*> Gal-
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diff changeset
3019 lileus quamquam vibrationes ten$arum chordarum differ&atilde;t
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diff changeset
3020 ab ijs pendulorum.</I>
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diff changeset
3021 <C><I>FINIS.</I></C>
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3022 <pb n=88>
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parents:
diff changeset
3023 <C>SPIEGATIONE</C>
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3024 <C>di vna nuoua $pecie di Bale$tra.</C>
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parents:
diff changeset
3025 <MARG><I>Tab.</I> 9.</MARG>
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diff changeset
3026 <p>IN que$ta figura $i e$prime vna nuoua inu&etilde;-
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parents:
diff changeset
3027 tione di Bale$tra, la quale, ma$$imamente
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parents:
diff changeset
3028 in grande, per tirar granate, &ograve; $a$$i pu&ograve; e$-
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diff changeset
3029 $ere di gran con$eguenze nella militare, co-
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parents:
diff changeset
3030 me dimo$trera$$i.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3031 <p>Dalle $ue parti $i verr&agrave; in cognitione del
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parents:
diff changeset
3032 modo di fabbricarla, e $ono le $eguenti.
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diff changeset
3033 <p>AM, MN $ono amendue le braccia. Il punto M &egrave; il cen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3034 tro della machina. Per la cauit&agrave; M deue pa$lar la pall&atail;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3035 $cagliata dalla corda; e per di $otto M $i ferma &amp; inca$lra
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3036 nel manico, al modo delle bale$tre communi. Ai due capi,
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diff changeset
3037 &ograve; $iano e$tremit&agrave; A, N $i annette la fune. I punti A, E, F,
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parents:
diff changeset
3038 G, I, K $ono in vna linea retta. Gl' interualli AE, EF, FG,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3039 GI, IK, $ono, benche non di nece$$it&agrave;, eguali. Le altezze,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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3040 &ograve; comme$$ure KL, IH, GD, FC, EB perpendiculari, nell'
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
3041 incuruar$i dell' arco, $i aprono intorno a' centri K, I, G, F,
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parents:
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3042 E. Donde ne $iegue, che prendendo la corda dal $uo mez-
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parents:
diff changeset
3043 zo, e tirandola ver$o O; amendue le braccia $i aprono nel-
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parents:
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3044 le predette comme$$ure, come compare nell' vno d'e$$i $e-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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3045 gnato a punti con le lettere corri$pondenti. Cia$cuna
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parents:
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3046 delle predette comme$$ure viene $trettamente rin$errat&atail;
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parents:
diff changeset
3047 da vna molla, come $i vede in L, H, D, C, B; e que$te mol-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3048 le, quanto pi&ugrave; $i auuicinano al centro M, deuono e$$ere pi&ugrave;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3049 grandi e pi&ugrave; ma$$iccie, in modo che, per cagione della gr&atilde;-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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3050 dezza opportuna, vengano ad aprir$i con egual facilit&agrave;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3051 dell'altre, e per cagione della gro$$ezza, habbiano nel $er-
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parents:
diff changeset
3052 rar$i maggior forza, &ograve; $ia momento, per la ragione, che $ot-
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parents:
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3053 to $i dir&agrave;.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3054 <p>Ci&ograve; pre$uppo$to, &egrave; facil co$a dimo$trare i vantaggi di
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parents:
diff changeset
3055 <pb n=89>
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parents:
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3056 que$ta machina $opra le ordinatie.
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parents:
diff changeset
3057 <p>Primieramente nel triangolo ALK, e$$endo le altezz&etail;
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parents:
diff changeset
3058 EB, FC, GD, IH, KL perpendicolari, e perci&ograve; paralelle;
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parents:
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3059 ne $iegue che le proportioni di AE ad EB, di AF ad FC, di
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parents:
diff changeset
3060 AG a GD, di AI ad IH, di AK a KL $ieno tutte eguali; e
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parents:
diff changeset
3061 douendo e$$ere parimente eguali le re$i$tenze delle molle
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parents:
diff changeset
3062 in B, C, D, H, L, che $i $uppongono di egual neruo nell'
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parents:
diff changeset
3063 aprir$i; ne $iegue ($econdo i principij della Meccanica) che
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parents:
diff changeset
3064 attraendo$i con la fune l'e$tremit&agrave; A, nel mede$imo tempo
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parents:
diff changeset
3065 e con la mede$ima facilit&agrave; vincera$$i l'equilibrio di tutte le
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parents:
diff changeset
3066 molle; la re$i$tenza delle quali $i con$idera in ragione di
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parents:
diff changeset
3067 pe$o, $i come le linee AE, EB; AF, FC; AG, GD; &amp;c. $i con-
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parents:
diff changeset
3068 $iderano come vetti, &ograve; lieue, che hanno i loro ippomoclij, &ograve;
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parents:
diff changeset
3069 $iano centri in E, F, G, I, K, e la potenza in A, la quale &egrave;
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parents:
diff changeset
3070 comune a tutte.
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parents:
diff changeset
3071 <p>In $econdo luogo, hauendo il braccio AE al braccio EB
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parents:
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3072 (il $imile dica$i degli altri) hauendo, dico, gran proportio-
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parents:
diff changeset
3073 ne, re$ter&agrave; molto ageuolato il moto.
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parents:
diff changeset
3074 <p>Terzo e$$endo molte le molle, e a prendo$i tutte, ne deue
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parents:
diff changeset
3075 $eguire vn notabile incuruamento d'amendue le braccia;
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parents:
diff changeset
3076 onde la$ciando l'arco in libert&agrave;, e chiudendo$i tutte le $u-
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parents:
diff changeset
3077 det te molle nel mede$imo tempo, cio&egrave; qua$i in vn'attimo;
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parents:
diff changeset
3078 dour&agrave; la corda, che era tirata ver$o O, pa$$are qua$i in i$t&atilde;-
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parents:
diff changeset
3079 te ver$o M; il che non potendo$i fare $e non con $omma ve-
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parents:
diff changeset
3080 locit&agrave;, per la grandezza dello $patio; e a que$ta corri$pon-
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parents:
diff changeset
3081 dendo la forza, ne $eguir&agrave; vn colpo molto con$iderabile, e
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parents:
diff changeset
3082 vantaggio$o, come cia$cuno pu&ograve; arguire.
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parents:
diff changeset
3083 <p>Re$tano hora a $eior$i alcune difficolt&agrave;. La prima &egrave;,
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diff changeset
3084 che, quantunque $ia vero, che quella forza ba$tante in A
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diff changeset
3085 per vincer l'equilibrio della molla B, quella mede$ima al-
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diff changeset
3086 tre$i $ia $ufficiente a vincer l' equilibrio di tutte l'altre, per
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diff changeset
3087 e$$ere eguali le proportioni delle vetti; ci&ograve; non o$tante, c&otilde;-
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diff changeset
3088 $iderando$i il braccio incuruato, come $i vede nell' arco
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diff changeset
3089 KLA $egnato a punti, le proportioni rie$cono alterate; do-
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diff changeset
3090 <pb n=90>
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diff changeset
3091 uendo$i prendere per le lunghezze delle vetti $udette, non
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parents:
diff changeset
3092 pi&ugrave; le lunghezze di prima, ma bensi le applicate di detto
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parents:
diff changeset
3093 arco, cioe af, ag, ai, ak; delle quali aK, e l' altre a lei pi&ugrave;
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parents:
diff changeset
3094 vicine $i abbreuiano molto pi&ugrave; quando l' arco &egrave; incurua-
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parents:
diff changeset
3095 to, che quando non &egrave;: Onde per tal ragione dourebbero
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parents:
diff changeset
3096 le parti pi&ugrave; vicine al centro M aprir$i meno dell'altre pi&ugrave;
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parents:
diff changeset
3097 vicine alle e$tremit&agrave;. A ci&ograve; $i ri$ponde, che per e$$er la
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parents:
diff changeset
3098 corda a o pi&ugrave; obliqua alla lunghezza a e di quel che $ia all'
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parents:
diff changeset
3099 altre pi&ugrave; vicine al centro M, quindi ne $iegue, che per quel'
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parents:
diff changeset
3100 altra cagione s' aprono pi&ugrave; ageuolmente le parti vicine al
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parents:
diff changeset
3101 centro; onde, temperata vna ragione con l' altra (quando
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parents:
diff changeset
3102 l' arco non $ia e$tremamente incuruato) $i con$egui$ce vno
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parents:
diff changeset
3103 $tato d'apertura opportuna.
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parents:
diff changeset
3104 <p>La $econda difficolt&agrave; &egrave; che cia$cuna molla nel $uo re-
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diff changeset
3105 $tringer$i, par che cagioni qualche effetto contrario all'in-
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diff changeset
3106 tento. Imperoche, per e$empio, nella molla B il mezzo
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diff changeset
3107 anello, che ri$guarda l'e$tremit&agrave; A, nello $tringer$i f&agrave; ben$i
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diff changeset
3108 il $uo douere, perche il $uo moto &egrave; ver$o il centro M; ma l'
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parents:
diff changeset
3109 altra met&agrave;, che ri$guarda il $udetto centro M, nello $trin-
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diff changeset
3110 ger$i, hauendo il $uo moto ver$o A, $i oppone al chiudi-
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parents:
diff changeset
3111 mento della molla $eguente C; e il $imile dica$i dell' altre.
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parents:
diff changeset
3112 A ci&ograve; $i &egrave; po$to rimedio col far pi&ugrave; grandi, e pi&ugrave; mafficcie
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diff changeset
3113 le molle pi&ugrave; vicine al centro M, accre$cendole, e ingro$$an-
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parents:
diff changeset
3114 dole di mano in mano opportunamente. Quindi ne $egue
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parents:
diff changeset
3115 che per la maggior grandezza c&otilde;$entono egualmente all'
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parents:
diff changeset
3116 aprir$i con facilit&agrave;; ma all' incontro nel $errar$i, per e$$ere
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parents:
diff changeset
3117 pi&ugrave; ma$$iccie, e di maggior corpo, vengono ad hauere
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3118 maggior momento delle men corpulenti, $uperando co&ntail;
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parents:
diff changeset
3119 ci&ograve; non $olo il detto moto oppo$to, ma etiandio impri-
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parents:
diff changeset
3120 mendo maggior moto al ferro dell'arco, con cui $i acco-
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parents:
diff changeset
3121 muna il moto.
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parents:
diff changeset
3122 <p>Auuerta$i, che quanto $aranno di maggior numero le
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3123 c&otilde;me$lure, le molle di maggior pe$o, e l'arco pi&ugrave; pouero di
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parents:
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3124 corpo, tanto riu$cir&agrave; il colpo a di$mi$ura maggiore, per l'
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parents:
diff changeset
3125 <pb n=91>
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parents:
diff changeset
3126 incuruamento notabile delle braccia, e per il maggior mo-
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parents:
diff changeset
3127 mento delle molle; e ci&ograve; con adoperare la mede$ima
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parents:
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3128 forza.
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parents:
diff changeset
3129 <p>Auuerta$i parimente, che il braccio AE, &egrave; il $uo corri$-
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parents:
diff changeset
3130 pondente deuono e$$ere alquanto pi&ugrave; corti, cio&egrave; A vna
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parents:
diff changeset
3131 delle e$tremit&agrave; dell'arco deue e$$ere pi&ugrave; ver$o il centro di
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parents:
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3132 quel che $ia il concor$o delle linee LB, KE, come pure dall'
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parents:
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3133 altra parte; perche $i vede che aprendo$i meno le parti vi-
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parents:
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3134 cine ad A, l'altre molle fanno miglior effetto.
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parents:
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3135 <p>Finalmente la $perienza ha mo$trato, che e$$endo$i la-
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parents:
diff changeset
3136 uorata vna tal machina con pochi$$imi nodi, ageuoli$$ima
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parents:
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3137 ad aprir$i, e $enza hauer ingrandite e ingro$late le molle,
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parents:
diff changeset
3138 che pi&ugrave; $i vanno auuicinando al centro M, come $i &egrave; det-
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parents:
diff changeset
3139 to; con tutto ci&ograve; l' ordigno &egrave; riu$cito di forza molto $upe-
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parents:
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3140 riore a vna bale$tra grande, e difficilif$ima a inarcar$i. On-
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parents:
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3141 de non dubito, che, facendo$i con tutte ie regole accenna-
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parents:
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3142 te, non debba riu$cire vna machina di effetto marauiglio$o
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parents:
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3143 aggiungendo che per tirar granate dourebbero i bracci
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parents:
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3144 e$$er di legno, armati di ferro $ol doue $i richiede.
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parents:
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3145 <pb n=92>
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parents:
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3146 <C>Nouum genus Bali$t&aelig;
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parents:
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3147 Explicatio.</C>
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parents:
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3148 <MARG><I>Tab.</I> 9.</MARG>
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parents:
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3149 <p>IN hac figura exprimitur nouum genus Bali-
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parents:
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3150 $t&aelig;, qu&aelig; machina pr&aelig;$ertim in mole maio-
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parents:
diff changeset
3151 ri, non parum vtilitatis afferre pote$t rei mi-
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parents:
diff changeset
3152 litari ad eiaculanda mi$$ilia, vt demon$tra-
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parents:
diff changeset
3153 bitur. Ex eius ver&ograve; partibus, quas $ubinde
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parents:
diff changeset
3154 recen$eo, etiam modus $tructur&aelig; apparebit.
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diff changeset
3155 <p>AM, MX $unt brachia. Punctum M centrum machi-
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parents:
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3156 n&aelig;. Per <*>auitatem M tran$it telum emi$$um. Infra M in-
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parents:
diff changeset
3157 $eritur manubrium, vt in bali$tis vulgaribus. Extremis
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parents:
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3158 capitibus A, N adnectitur funis. Puncta A, E, F, G, I, K
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3159 $unt in linea recta. Interualla AE, EF, FG, GI, IK $unt (li-
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parents:
diff changeset
3160 c&egrave;t non nece$$ari&ograve;) &aelig;qualia. Altitudinis, $eu commi$$ur&aelig;
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parents:
diff changeset
3161 KL, IH, GD, FC, EB $unt perpendiculares rect&aelig; occult&aelig;
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parents:
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3162 KA. Singul&aelig; autem, dum curuatur arcus, aperiuntur cer-
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parents:
diff changeset
3163 ca centra K, I, G, F, E. Hinc $equitur vt funis ex medio
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parents:
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3164 dum attrahitur in O, aperiantur pr&aelig;dict&aelig; commi$$ur&aelig;, $eu
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parents:
diff changeset
3165 nodi, &amp; curuentur vtraque brachia, vt in eorum altero ap-
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parents:
diff changeset
3166 paret punctis notato. Quilibet ex his nodis arcti$lim&egrave; $trin-
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parents:
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3167 gitur $upern&egrave;, a $uo elaterio, vt videre e$t in L, H, D, C, B.
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parents:
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3168 Elateria autem qu&ograve; propinquiora centro M tanto maiora,
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parents:
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3169 &amp; cra$$iora debent e$$e remotioribus: Hinc fit vt, propter
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parents:
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3170 molem opportun&egrave; auctam, &aelig;qu&egrave; facil&egrave; aperiantur, ac c&aelig;-
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parents:
diff changeset
3171 tera; &amp; vice ver$a, propter cra$$itiem maiorem, $ibi relicta
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3172 validi&ugrave;s re$tringantur. Cuius rei paulo infra rationem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3173 dabimus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3174 <p>His po$itis facile e$t o$tendere, quantum pr&aelig;$tet hu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3175 iu$cemodi machina vulgaribus &amp; communibus bali$tis.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3176 <p>Prim&ugrave;m, in Triangulo ALK c&ugrave;m altitudines EB, FC,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3177 GD, IH, KL $int perpendiculares, ideoque parallel&aelig;, hinc
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3178 <pb n=93>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3179 $it vtrationes AE ad EB, AF ad FC, AG ad GD, AI ad
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3180 IH, AK ad KL $int &aelig;quales. Sunt pariter &aelig;quales re$i-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3181 $tenti&aelig; elateriorum in B, C, D, H, L (po$uimus enim ela-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3182 teria ita opportun&egrave; aucta vt &aelig;qu&egrave; facile $ingula aperian-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3183 tur) ergo (ex primis principijs mecanicorum) dum attra-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3184 huntur fune extrema capita A, N, eodem tempore, eadem-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3185 que facili ate vincetur &aelig;quilibrium omnium elateriorum,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3186 quorum re$i$tentia in $ingulis con$ideratur in ratione pon-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3187 deris, quemadmodum line&aelig; AE, EB; AF, FC; AG, GD
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3188 &amp;c. con$iderantur vt vectes, quorum hippomoclia $eu c&etilde;-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3189 tra $unt in E, F, G, I, K, potentia autem con$ideratur in A
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3190 communis omnibus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3191 <p>Scund&ograve;, c&ugrave;m AE ad EB (idem die de c&aelig;teris) habeant
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3192 magnam proportionem, facil&egrave; aperientur nodi, &amp; curuabi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3193 tur arcus; quantumuis augeatur numerus nodorum.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3194 <p>Terti&ograve; Cum $int plures nodi, atque omnes aperiantur,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3195 nece$$e e$t vt brachia arcus vald&egrave; incuruentur; quamobr&etilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3196 $i idem arcus $ibi relinquatur, pr&aelig;dicti nodi omnes, vi ela-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3197 teriorum, ictu oculi claudentur; e odemque puncto tempo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3198 ris corda ex O percurret totum $patium v$que ad M: Qu&ograve;d
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3199 c&ugrave;m fieri nequeat ni$i $umma velocitate, propter magni-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3200 tudinem pr&aelig;dicti $patij, &amp; velocitati re$pondent vis, atque
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3201 impetus, nece$$e e$t vt hinc $equatur ictus valde notabilis,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3202 vt facil&egrave; e$t vnicuique conijcere.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3203 <p>Super $unt nunc difficultates nonnull&aelig; $oluend&aelig;. Prima
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3204 e$t, qu&ograve;d lic&egrave;t vis $ufficiens in A ad vincendum &aelig;quilibri&utilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3205 elaterij B, illa eadem quoque $ufficiat ad vincendum &aelig;qui-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3206 librium c&aelig;terorum, propter &aelig;quales proportiones vecti&utilde;;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3207 his tamen non ob$tantibus, $i con$ideretur brachium iam
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3208 incuruatum, vt apparet in KLA punctis notato, proportio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3209 nes ill&aelig; cernuntur notabiliter variat&aelig;. Neque enim pro
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3210 longitudinibus vectium $umi po$$unt longitudines priores,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3211 $ed loco ip$arum accipiend&aelig; $unt applicat&aelig; arcus, videli-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3212 cet af, ag, ai, ak quarum ak, eidemque propinquiores, qu&atilde;-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3213 <pb n=94>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3214 do arcus incuruatur, breuiores fiunt, qu&agrave;m e$$ent ante&atail;.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3215 Re$pondeo, qu&ograve;d corda ao c&ugrave;m $it obliquior re$pectu
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3216 longitudinis ae, qu&agrave;m re$pectu c&aelig;terarum centro propin-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3217 quiorum, hinc fit vt, quant&ugrave;m e$t ex hac ratione, facili&ugrave;s
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3218 aperiantur partes propinquiores centro; quamobrem, vtra-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3219 que ratione inuicem temperata, dummodo arcus non $it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3220 $umm&egrave; incuruatus omnes partes aperientur, quantum $a-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3221 tis e$t ad intentum.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3222 <p>Altera difficultas e$t, quod elaterium quodlibet dum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3223 re$tringitur videtur ob$tare motui elaterij $equentis. Nam,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3224 exempli gratia, in elaterio B $emiannulus re$piciens extre-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3225 mum A, dum $tringitur, optim &egrave; pr&aelig;$tat $uum effectum, c&utilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3226 eius motus $it vers&ugrave;s centrum M At &egrave; contrario reliqua
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3227 pars, $eu $emiannulus re$piciens pr&aelig;dictum centrum M, c&utilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3228 habeat $uum motum ver$us A videtur ob$tare, quo minus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3229 liber&egrave; claudatur $equens elaterium C. Aque idem de c&aelig;te-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3230 ris dicendum. Huic incommodo con$ultum e$t augendo
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3231 magnitudinem, &amp; cra$$itiem elateriorum, qu&ograve; magis acce-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3232 dunt ad centrum M. Hinc enim $equitur vt propter ma-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3233 gnitudinem facil&egrave; con$entiant arcui, dum incuruatur; at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3234 dum idem arcus liber&egrave; $ibi relinquitur, cum $int corpulen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3235 tiora &amp; cra$$iora habent maius momentum, qu&agrave;m c&aelig;tera
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3236 graciliora, ideoque non modo vincunt motum illum op-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3237 po$itum, $ed etiam imprimunt maiorem motum ferro ar-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3238 cus, cui ille motus communicatur.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3239 <p>Aduerte quod commi$$ur&aelig; $eu nodi, qu&ograve; plures fuerint,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3240 elateria autem maioris ponderis, arcus denique corporis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3241 gracilioris equ&aelig; expeditioris, tanto ictus longat pr&aelig;$tan-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3242 tior $equetur, tum propter notabilem curuaturam brachio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3243 rum, tum propter momentum maius elateriorum, &amp; quid&etilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3244 po$ita eadem potentia, aut etiam minori, pro vt longitudi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3245 nes vectium $tatuuntur.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3246 <p>Aduerte etiam, longitudinem brachij AE, eiu$demqu&etail;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3247 corre$pondentis debere c&aelig;teris paribus nonnihil imminui,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3248 <pb n=95>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3249 quod fiet $i A, alterum extremum arcus, $it propri&ugrave;s cen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3250 tro M, qu&agrave;m $it concur$us linearum LB, KE. Idem dicen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3251 dum de altero extremo N. Nam c&ugrave;m minus aperiantur
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3252 partes propinquiores puncto A, c&aelig;tera elateria, vt com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3253 pertum e$t, meliorem effectum pr&aelig;$tant.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3254 <p>Fauet denique experientia. Nam huiu$cemodi machi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3255 na pauci$$imis nodis con$tructa, facillim&aelig; curuatur&aelig;, cum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3256 elaterijs eiu$dem pror$us molis &amp; cra$$itudinis; nihilomi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3257 nus long&egrave; $uperauit vim bali$t&aelig; communis maxim&aelig;, &amp; dif
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3258 ficillim&aelig; flexionis. Quamobrem non dubito quin, $i pr&aelig;-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3259 cepta $uperi&ugrave;s data exact&egrave; $eruentur, elaborari po$$it ma-
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3260 china mir&aelig; vtilitatis. Adde po$tremo ad iacienda qu&aelig;d&atilde;
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3261 mi$$ilia, vt e$t genus quoddam bolidum, vulgo <I>granate,</I> op-
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3262 portuniora e$$e brachia lignea, tantummodo, vbi nece$$i-
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3263 tas po$tulat, armata ferro.
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3264 <C><I>FINIS.</I></C>
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3265 <pb>
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3266 <p><I>Vid. D. Bernardus Marchellus Re-
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3267 ctor P&oelig;nitent. in Metropol. Bonon.
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3268 pro Illu$tri$s. &amp; Reverendi$s. Domino
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3269 D. Iacobo Boncompagno Archiepi$-
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3270 copo, &amp; Principe.</I>
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3271 <p><I>Vid. Silue$ter Bonfiliolus Inqui$itionis
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3272 reui$or, &amp; imprimi po$$e cen$uit.</I>
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3273 <p><I>Stante atte$tatione.</I>
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3274 <C><I>Imprimatur.</I></C>
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3275 <p><I>F. Io$eph Maria Agudius Vicarius
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3276 Sancti Offi c ij Bononi&aelig;.</I>
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3277 <C>8 00 57</C>
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3278 <pb>
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3279 <CAP>BVLA I.</CAP>
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3280 <FIG>
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3281 <pb>
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3282 <CAP>TABVLA I.</CAP>
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3283 <FIG>
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3284 <pb>
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3285 <CAP>TABVLA II.</CAP>
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3286 <FIG>
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3287 <pb>
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3288 <CAP>TABVLA III.</CAP>
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3289 <FIG>
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3290 <pb>
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3291 <CAP>TABVLA VI.</CAP>
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3292 <FIG>
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3293 <pb>
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3294 <CAP>TABVLA V.</CAP>
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3295 <FIG>
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3296 <pb>
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3297 <CAP>TABVLA IV.</CAP>
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3298 <FIG>
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3299 <pb>
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3300 <CAP>TABVLA VII.</CAP>
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3301 <FIG>
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3302 <pb>
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3303 <CAP>TABVLA VIII.</CAP>
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3304 <FIG>
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3305 <pb>
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3306 <FIG>
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3307 <pb>
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3308 <FIG>
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3309 <CAP>TABVLA VIIII.</CAP>