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annotate texts/archimedesOldCVSRepository/archimedes/raw/stell_mecha_01_it_1597.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
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1 <pb id="p.0001">
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2 <HEAD>DE GLI
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3 ELEMENTI
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4 MECHANICI</HEAD>
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5 <FIG>
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6 La $tatera. Leua.
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7 Raggi nell'a$$e. Rote vettiue
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8 Taglia. Rote motiue.
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9 Cugno. Vite.
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10 <HEAD><I>DI C. ANTONIO STELLIOLA.</I></HEAD>
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11 <FIG>
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12 <HEAD>IN NAPOLI, Nella Stamparia &agrave; Porta Regale
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13 M. D. XCVII.</HEAD>
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14 <p n=>1</p>
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15 <HEAD><I>PROPOSITIONE
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16 di tutta l'opera.</I></HEAD>
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17 <p>Cerchiamo come po$$a la potenza
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18 minore vincer di forza la maggiore:
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19 e la potenza piu tarda, vincer di mo-
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20 uimento la piu veloce. e que$to con
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21 Leue, Taglie, Viti, Rote, e tutti in$trumenti che
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22 moltiplicar po$$ono il momento, o della forza,
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23 o della velocit&agrave;. Qual $oggetto communemente
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24 gli antichi chiamarono Mechaniche. Il che tut-
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25 to $i tratter&agrave; $econdo le $uppo$itioni fatte de mo-
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26 menti, o per linee parallelle, o per linee con-
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27 correnti ad vn ponto, o per circonferenze d'in-
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28 torno vn centro i$te$$o: e $econdo il $olito v$o de
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29 mathematici deducendo le dimo$trationi, e cau&shy;
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30 $e de gli effetti, dalli primi e proprij principij.
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31 <FIG>
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32 <foot>A</foot>
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33 <p n=>2</p>
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34 <HEAD><I>DEFINITIONI.</I>
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35 I.</HEAD>
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36 <p>Centro di pe$o diciamo il ponto, per cui il corpo co-
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37 munque $o$pe$o, non muta po$itione.
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38 <HEAD>II.</HEAD>
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39 <p>Corpo egualmente di$te$o diciamo, che comunque
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40 tagliato con pianezze parallele, fa figure $uperficiali
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41 eguali e $imili.
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42 <HEAD>III.</HEAD>
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43 <p>Applicar$i diciamo vn corpo ad vna linea, quando
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44 detto corpo vgualmente di$te$o occupi la lunghezza di
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45 detta linea.
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46 <HEAD>IIII.</HEAD>
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47 <p>Linea di momento diciamo, per cui il centro di pe-
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48 $o della grauezza da impedimento libera $i moue.
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49 <HEAD>V.</HEAD>
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50 <p>Libra &ograve; $tatera diciamo la linea a cui $i applicano, &ograve;
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51 appendono le grauezze: e che $ia $u$pe$a da vn $ol
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52 ponto.
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53 <HEAD>VI.</HEAD>
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54 <p>E leua diciamo la linea $o$tenuta da due ponti, o $o-
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55 $tenuta da vn ponto e mo$$a da vna po$$anza.
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56 <HEAD>VII.</HEAD>
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57 <p>Ponto di momento diciamo nella $tatera e leua, il
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58 ponto, nel quale s'incontra la linea del momento, con
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59 la linea della $tatera.
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60 <HEAD>VIII.</HEAD>
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61 <p>E ponto di appen$ione: il ponto, onde perde la gra-
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62 <p n=>3</p>
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63 uezza $taccata dalla $tatera, o leua, nelquale i$te$$o pon-
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64 to s'intende hauer il $uo momento.
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65 <HEAD>IX.</HEAD>
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66 <p>Et Horizonte de pe$i la $uperficie in cui le linee de
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67 momenti tutte vanno perpendicolarmente.
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68 <HEAD><I>Appendice.</I></HEAD>
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69 <p>Dalche &egrave; manife$to, che l'Horizonte de' momenti pa-
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70 ralleli, $ia $uperficie piana: e delli concorrenti $ia $uper-
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71 cie sferica.
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72 <HEAD><I>POSITIONI.</I>
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73 I.</HEAD>
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74 <p>Pigliamo nelli corpi egualmente di$te$i il centro del
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75 pe$o e$$er nella $uperficie, che diuide egualmente la
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76 lunghezza di detto corpo.
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77 <HEAD>II.</HEAD>
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78 <p>Che grauezze eguali appe$e o nell'i$te$$o ponto, o in
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79 ponti della libra egualmente di$tanti dalla $u$pen$ione
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80 della $tatera, habbiano momento eguale.
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81 <HEAD>III.</HEAD>
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82 <p>Che nelli corpi di vna i$te$$a natura $ia proportionale
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83 il pe$o alla quantit&agrave; delli corpi.
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84 <HEAD>IIII.</HEAD>
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85 <p>E, che la grauezza appe$a non $i fermi, $in che il c&etilde;-
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86 tro del pe$o non $ia nella perpendicolare del ponto del
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87 $o$tenimento.
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88 <foot>A 2</foot>
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89 <p n=>4</p>
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90 <HEAD><I>PROPOSITIONE.</I>
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91 I.</HEAD>
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92 <p>Se $i togliono due quantit&agrave; da due altre, che $iano
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93 eguali, e tra di loro, &amp; alla compo$ta delle due tolte: di-
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94 co che le re$tanti alle tolte $cambieuolmente $ono egua&shy;
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95 li.
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96 <FIG>
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97 <HEAD><I>Dimo$tratione.</I></HEAD>
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98 <p><I>Siano le due quantit&agrave;, A &amp; B, &amp; alla compo$ta di ambe $iano egua
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99 li, la C D, &amp; la E F; e dalla C D, toglia$i eguale ad A, che $ia,
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100 C G, e dalla E F toglia$i eguale a B, che $ia E H. dico che la re$tan-
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101 te H F, &egrave; vguale ad A; e la G D, eguale a B. Si mo$tra perci&ograve;
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102 che e$$endo C D, eguale ad A e B in$ieme: tolti dall'vna e l'altra $um-
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103 male A, e C G eguali: le re$tanti, B, e G D di con$eguenza $o-
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104 no eguali. Similmente perche la E F $i pone vguale alle A, &amp; B
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105 gionte in$ieme; tolte la E H, &amp; B vguali: le re$tanti, H F, e A $o-
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106 no di con$eguenza eguali. &egrave; adunque la H F eguale a C G: e la G D
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107 eguale ad E H. il che hauea da mo$trar$i.</I>
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108 <HEAD><I>Appendice.</I></HEAD>
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109 <p>Dalche &egrave; manife$to, che le i$te$$e re$tanti $cambieuol&shy;
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110 mente $ono proportionali alle tolte.
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111 <p n=>5</p>
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112 <p>Percioche e$$endo le C G H F eguali. e le G D E H
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113 anco eguali: ma le eguali $ono proportionali: $ono d&utilde;que
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114 come C G ad E H, co$i H F ad G D: ilche hauea da mo-
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115 $trar$i.
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116 <HEAD><I>PROPOSITIONE.</I>
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117 II.</HEAD>
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118 <p>Se alla linea della $tatera $i applicano continuatamen-
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119 te due corpi: li centri delli corpi applicati, $ono di$tanti
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120 dal centro ditutto il compo$to, di di$tanze proportio-
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121 nali alli pe$i, pigliati reciprocamente.
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122 <FIG>
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123 <HEAD><I>Dimo$tratione.</I></HEAD>
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124 <p><I>Sia la linea della $tatera, A B, l'vn delli corpi applicati $ia
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125 B C, l'altro $ia C A, e l'applicatione del corpo B C occupi la parte di li-
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126 nea B D, e del corpo C A, la parte A D, e diuida$i B D, in par-
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127 ti vguali al ponto E: &amp; A D in parti eguali il ponto F: &egrave; manifesto
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128 che del corpo applicato &agrave; B D, il ponto del momento $ia E, e del cor-
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129 po applicato a D A, il ponto del momento $ia F, dico che diui$a B A
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130 tutta per met&agrave; nel ponto G, che &egrave; ponto di mom&etilde;to
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131 della grauezza tut&shy;
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132 ta compo$ta di ambedue: c'habbia la di$tanza F G a G E la ragione
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133 che'lpe$o di B C al pe$o di C A. Si mo$tra percioche la ragione del
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134 pe$o di B C, al pe$o di C A, e l'i$te$$a che delli corpi: e delli corpi</I>
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135 <p n=>6</p>
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136 <I>vgualmente di$te$i, e l'i$te$$a che delle linee: qual &egrave; della linea B D
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137 a D A. e delle loro met&agrave; di E D a D F cio&egrave; di F G, a G E:
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138 &amp; perche $e due quantit&agrave; compongono quantit&agrave;, e le met&agrave; del-
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139 le componenti, compongono la met&agrave; della tutta: ma le met&agrave; delle li-
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140 nee componenti $ono A F, e B E, la met&agrave; della tutta, e co$i la B G
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141 come la A G. perci&ograve; togliendo due quantit&agrave; A F B E dalle due,
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142 A G, G E eguali tra di loro, &amp; alla compo$ta di A F, B E.
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143 le re$tanti $cambieuolmente $ono proportionali, e perci&ograve; F G, a G E
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144 $ar&agrave; nell'i$te$$a ragione di B E, ad A F, cio&egrave; della doppia, B D, a D
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145 A: qual &egrave; l'i$te$$a del corpo, B C, a C A: e della grauezza di B C, a
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146 C A. la di$tanza dunque F G, alla di$tanza E G, ha la ragione che'l
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147 pe$o di B C, al pe$o di C A, il che $i hauea da mo$trare.</I>
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148 <HEAD><I>PROPOSITIONE.</I>
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149 III.</HEAD>
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150 <p>Se ad vn vna $tatera $iano appe$e due grauezze, e l'
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151 interuallo delli ponti della $o$pen$ione $i diuida nella
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152 ragione delle grauezze: $o$pe$a la $tateradal ponto del-
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153 la diui$ione, $ta in equilibrio.
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154 <FIG>
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155 <HEAD><I>Dimo$tratione.</I></HEAD>
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156 <p><I>Sia la $tatera A B: le grauezze in e$$a $o$pe$e: C, &amp; D: la C, dal
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157 ponto, A, &amp; la D, dal ponto, &lsquo;B, &amp; in quellaragione che ha la gra-
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158 uezza D, alla grauezza C, $i diuida A B nelponto E dico che $o$pe$a</I>
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159 <p n=>7</p>
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160 <I>la $tatera nel ponto E, $ta in equilibrio. Si mo$tra alla linea, B E, ta-
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161 gli$i eguale la linea A F, dunque giunta communemente, F E, $ar&agrave; B
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162 F, vguale ad A E, e perci&ograve; haur&agrave; B F, ad F A, l'i$te$$a ragione, che
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163 D, a C. faccia$i alla B F, vguale, B G, &amp; alla A F, vguale A H,
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164 dunque $e alla linea, G F, s'intenda applicato un corpo eguale di pe$o
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165 alla grauezza D, e tal corpo $i allunghi nella iste$$a gro$$ezza fin ad H,
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166 $ar&agrave; il corpo applicato ad F H, uguale di pe$o a C: percio che hauendo
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167 G F, ad F H, l'i$te$$a ragione che D, a C, e li corpi applicati l'i$te$-
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168 $a delle linee: $ono perci&ograve; come la grauezza D alla C, co$i il cor-
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169 po applicato ad F G, al corpo applicato ad H F: dunque mutando, $o-
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170 no anco proportionali: ma il corpo applicato a G F, &egrave; di pe$o uguale al
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171 D, dunque l'applicato ad F H &egrave; vguale di pe$o a C: &amp; &egrave; delli due ap-
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172 plicati, il commune punto di momento in E. Dunque delli D C in$ie-
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173 me pigliati il commun momento &egrave; nel ponto i$te$$o: &amp; percio la $tatera
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174 $o$tenuta in E, $ta in equilibrio, ilche $i hauea da mo$trare.</I>
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175 <HEAD><I>Appendice.</I> I.</HEAD>
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176 <p>Dal che &egrave; manife$to che'l centro commune di due
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177 pe$i &egrave; il ponto che diuide l'interuallo de'centri loro, re-
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178 ciprocamente.
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179 <HEAD><I>Appendice.</I> II.</HEAD>
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180 <p>E $e due grauezze diui$amente $i appendono: che di-
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181 ui$o l'interuallo nella ragione delle grauezze recipro-
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182 camente: dette grauezze, fanno l'i$te$$o effetto nel mo&shy;
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183 m&etilde;to, che $e in detto ponto giuntam&etilde;te fu$$ero appe$e.
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184 <HEAD><I>PROPOSITIONE.</I>
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185 IIII.</HEAD>
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186 <p>Se due grauezze appe$e in due ponti facciano equi-
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187 pondio: e di nuouo appe$e in due altri ponti facciano
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188 <p n=>8</p>
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189 equipondio; l'interualli delle $o$pen$ioni mutate, $ono
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190 proportionali con li pe$i reciprocamente.
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191 <FIG>
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192 <HEAD><I>Dimo$tratione.</I></HEAD>
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193 <p><I>Sia la $tatera A B: il ponto della $o$pen$ione C, li ponti onde $one
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194 appe$e le grauezze che fanno equipondio A &amp; B. le grauezze appe$e
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195 D &amp; E. Quali di nuouo appe$e nelli ponti F &amp; G faccino equipondio:
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196 dico che la F A interuallo delle due $o$pen$ioni di D, a B G, inter-
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197 uallo delle $u$p&etilde;$ioni di E; ha quella ragione che la grauezza c alla gra-
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198 uezza D. Si mo$tra perche D et E grauezze nella $u$p&etilde;$ion prima han-
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199 no equipondio: dunque la ragione della grauezza D ad E, &egrave; l'i$te$$a che
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200 di B C a C A: e nella $econda $u$pen$ione la ragione di D ad E e l'i$te$-
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201 $a che di G C a C F. e perci&ograve; come B C &agrave; C A, co$i G C &agrave; C F, e per che
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202 da due $i togliono due altre nell'i$te$$a ragione, le re$tanti anco $ono nel-
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203 l'i$te$$a ragione. &egrave; dunque B G ad F A, come D ad E, ilche hauea da
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204 mo$trar$i.</I>
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205 <HEAD><I>PROPOSITIONE.</I>
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206 V.</HEAD>
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207 <p>Se due grauezze facciano equipondio, e gionte &ograve; tol-
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208 te due altre grauezze facciano anco equipondio: le gion&shy;
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209 te ancora &ograve; le tolte $ono nell'i$te$$a raggione.
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210 <p n=>9</p>
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211 <FIG>
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212 <HEAD><I>Dimo$tratione.</I></HEAD>
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213 <p><I>Sia la $tatera A B: il ponto della $u$pen$ione C: le grauezze appe-
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214 $e D et E: che facciano equipondio: e di nouo aggiuntoui due altre F e G
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215 facciano anco equipondio. dico che la grauezza F a G, ha la ragione
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216 che D ad E: qual'&egrave; l'i$te$$a che di B C a C. A. $i mo$tra perche D &amp; E,
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217 fanno equipondio. &amp; F e G fanno equipondio: perci&ograve; $ar&agrave;, come B C
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218 &agrave; C A co$i D F ad E G e nell'i$te$$a era D ad E dunque le re$tanti F e G
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219 $ono anco nell'i$te$$a ragione: e non altrimente che nella $uppo$ition del
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220 la compo$ta, $i mo$tra nella $uppo$ition delli re$idui. Ha$$i dunque l'in-
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221 tento.</I>
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222 <HEAD><I>PROPOSITIONE.</I>
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223 VI.</HEAD>
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224 <p>Date quante $i voglia grauezze appe$e in vn'i$te$$a
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225 $tatera, ritrouare il ponto del momento commune.
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226 <HEAD><I>Dimo$tratione.</I></HEAD>
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227 <p><I>Sia la $tatera A B dalli cui ponti A e B $iano $o$pe$e le grauezze C D
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228 e $iano in altri ponti $o$pe$i altri pe$i, come E nel ponto F: $i cerca il pon&shy;
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229 to del momento commune. diuida$i la B A nella ragione di C a D reci-
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230 procamente $e dunque il detto punto uiene in F e$$endo F il ponto del</I>
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231 <foot><I>B</I></foot>
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232 <p n=>10</p>
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233 <FIG>
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234 <I>momento delle C D pigliate in$ieme, $ar&agrave; ponto di momento commu
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235 ne delle grauezze C E D, tutte. Et harra$$i l'intento.</I>
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236 <p><I>Ma $e'l dato ponto ca$chi altroue come in H, perche le grauezze
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237 D, &amp; C appe$e in A e B fanno l'i$te$$o effetto che $e giuntamente fu$$e&shy;
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238 ro appe$e in H: perci&ograve; $e quella ragione che h&agrave; il compo$to di C D
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239 ad E habbia reciprocamente F G a G H, $ar&agrave; G ponto di momento
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240 commune di tutti. con l'i$te$$o ordine $i ritrouer&agrave; il centro di quante
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241 altre $i uogliano, il che $i hauea da trouare.</I>
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242 <HEAD><I>PROPOSITIONE</I>
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243 VII.</HEAD>
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244 <p>Delle grauezze che fanno equip&otilde;dio, compo$te le ra&shy;
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245 gioni delle grauezze e delle di$tanze, li e$tremi termini
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246 $ono eguali.
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247 <HEAD><I>Dimo$tratione.</I></HEAD>
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248 <p><I>Sia la $tatera A B il ponto del $ostenimento C le due grauezze che
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249 fanno equip&otilde;dio D &amp; E: de quali la D $ia $o$pe$a dal ponto A la E dal p&otilde;
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250 to B: dico che compo$tala ragione della grauezza D ad E: e della di$t&atilde;za
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251 A C a C B: cio&egrave; fatto che la qu&atilde;tit&agrave; F a G $ia come la grauezza D ad E e
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252 la quantit&agrave; G ad H come la di$tanza A C alla C B, che F &amp; H</I>
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253 <p n=>11</p>
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254 <FIG>
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255 <I>e$tremi termini $iano uguali. $i mo$tra: perche A C a C B $i &egrave; po$ta co-
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256 me G ad H: dunque riuoltando H &agrave; G, &egrave; come B C &agrave; C A. e per l'e-
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257 quipondio, come la di$tanza B C a C A co$i la grauezza D ad E, &amp;
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258 come D ad E co$i $i &egrave; pigliato F a G: dnnque F a G e come B C a C A,
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259 e nell'i$te$$a ragione era H a G. hanno dunque li due termini F et H l'i-
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260 $te$$a ragione al termine G. e perci&ograve; li F &amp; H $ono eguali tra di loro:
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261 il che $i hauea da mo$trare.</I>
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262 <HEAD><I>PROPOSITIONE</I>
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263 VIII.</HEAD>
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264 <p>Li momenti delle grauezze uguali, appe$e in di$tan&shy;
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265 ze ineguali, hanno fra di loro la proportione che le di-
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266 $tanze.
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267 <HEAD><I>Dimo$tratione.</I></HEAD>
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268 <p><I>Sia la $tatera A B, il ponto del $o$tenimento C, le grauezze uguali D
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269 &amp; E. de quali il D $ia appe$o in A, &amp; l'c in F. dico che il momento
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270 di D al momento di E, h&agrave; quella ragione che l'interuallo di A C all'-
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271 interuallo di F C. $i mo$tra, pigliato dall' altra parte del $e$tem-
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272 mento C, qual $i uoglia ponto B: intenda$i in e$$a appe$e due grauzze,</I>
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273 <foot><I>B</I> 2</foot>
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274 <p n=>12</p>
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275 <FIG>
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276 <I>vna che faccia equip&omacr;dio a D &amp; $ia G: et vn'altra che faccia equip&omacr;dio
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277 ad E. &amp; $ia H. perche dunque G a D ha quella ragione che A C a C B
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278 &amp; D ouero E ad H, hala ragione di B C a C F. dunque di pari il pri-
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279 mo termine A C all'ultimo F C, ha quella ragione, che il primo ter-
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280 mine G, al terzo H. $e dunque G ad H hal'i$te$$a ragione che la di$tan&shy;
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281 za A C alla di$tanza F C: &amp; il momento di G &egrave; uguale al momento
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282 di D appe$o in A, &amp; il momento di H vguale al momento di E appe$o
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283 in F. dunque il momento di D al momento di E ha quella ragione che
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284 la di$tanza A C alla di$tanza F C. il che $i bauea da mo$trare.</I>
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285 <HEAD><I>PROPOSITIONE.</I>
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286 IX.</HEAD>
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287 <p>Li momenti delle grauezze $o$pe$e in qual $i uoglia
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288 ponti della $tatera, han tra di loro la ragion compo$ta,
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289 della ragion delle grauezze, e delle di$tanze.
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290 <HEAD><I>Dimostratione.</I></HEAD>
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291 <p><I>Sia la $tatera A B il p&omacr;to del $o$tenim&etilde;to C le grauezze appe$e D dal
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292 p&omacr;to A, &amp; E dal ponto F. dico che la ragione del mom&etilde;to D al mom&etilde;
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293 to E, e compo$ta di due ragioni cio&egrave; della ragione della grauezza D
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294 alla grauezza E, e della di$tanza di A C alla F C. $i mo$tra appenda
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295 $i da B la grauezza G che faccia equipondio. a D, &amp; il pe$o H che fac&shy;</I>
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296 <p n=>13</p>
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297 <FIG>
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298 <I>cia equip&omacr;dio all' E: dico prima che la grauezza G alla grauezza H G ha
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299 la ragion compo$ta, di D ad E, e di A C ad F C. per il che da mo$trare: in|-
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300 tenda$i nell' A $o$pe$a la grauezza I uguale alla grauezza E, &egrave; manife$to
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301 che'l momento I al momento E, h&agrave; quella ragione che l'interuallo
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302 A C all'interuallo F C come nel pa$$ato habbiamo mo$trato: &amp; il mo-
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303 mento di D al momento d'I h&agrave; la ragione che la grauezza D alla gra-
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304 uezza I: perche $ono da un'i$te$$o ponto $o$pe$i. e$$endo dunque tre ter
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305 mini in continua habitudine il momento D, il momento I, &amp; il mom&etilde;&shy;
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306 to E: la ragione del primo termine al terzo &egrave; compo$ta della ragione
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307 di primo a $econdo e della ragione di $econdo a terzo: ma di primo
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308 a $econdo &egrave; di grauezza a grauezza: di$econdo a terzo &egrave; d'interuallo
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309 ad'interuallo. dunque, la ragione delli mom&etilde;ti di D ad E, che &egrave; l'i$te$$a
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310 che della portione G alla portione H: &egrave; compo$ta della ragione delle
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311 grauezze e della ragione delle di$tanze. Il che $i hauea da mo$trare.</I>
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312 <HEAD><I>PROPOSITIONE.</I>
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313 X.</HEAD>
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314 <p>Data qual $i uoglia grauezza, e li ponti della $o$pen|-
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315 $ion della $tatera, e della grauezza: e dato il pe$o del
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316 marco, ritrouare il luogo, oue detto marco faccia e-
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317 quipondio con la grauezza data.
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318 <p n=>14</p>
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319 <FIG>
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320 <HEAD><I>Dimo$tratione.</I></HEAD>
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321 <p><I>Sia la $tatera A B il ponto della $o$pen$ione C, la grauezza data D,
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322 qual poniamo che $i $o$penda in A: il marco dato di pe$o E: $i cerca il
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323 ponto oue detto marco appe$o faccia equipondio. per que$to: faccia$i
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324 che quella ragione che h&agrave; il pe$o E al pe$o D, quella habbia la linea A
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325 C a C F, dico che appe$o il marco in F fa equipondio, cio&egrave; che'l pon|-
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326 to del momento commune delle grauezze D &amp; e$ia il ponto della $o-
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327 $pen$ione C: il che &egrave; manife$to, percioche $ono li pe$i reciprochi al-
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328 le di$tanze. Ha$$i dunque l'intento.</I>
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329 <HEAD><I>PROPOSITIONE.</I>
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330 XI.</HEAD>
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331 <p>Data una $tatera, a cui $ia ugualmente applicato un
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332 corpo, e data una grauezza $o$pe$a da un dato ponto,
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333 e dato il pe$o del marco, ritrouare il ponto onde detto
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334 marco $o$pe$o faccia equipondio con la grauezza.
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335 <HEAD><I>Dimo$tratione.</I></HEAD>
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336 <p><I>Sia la $tatera A B il ponto del $o$tenimento C, il corpo applicato
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337 A D E B, la grauezza $o$pe$a H I; il ponto onde la grauezza &egrave; appe-
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338 $a in A: il marco L, $i cerca il ponto onde $o$pe$o il marco, faccia e-
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339 quipondio con H I, Per que$to: faccia$i alla linea A C uguale la C F,</I>
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340 <p n=>15</p>
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341 <FIG>
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342 <I>dunque il corpo D F applicato ad A F $t&agrave; in equilibrio nel ponto della
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343 $o$pen$ione C. et diui$a F B re$tante per met&agrave; nel ponto G: del re$tan-
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344 te corpo F E applicato alla linea F B, $ar&agrave; G, il ponto di momento. $e
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345 dunque la ragione che h&agrave; C F ad F G, habbia la grauezza E F alla
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346 parte del pe$o I, $tar&agrave; il corpo F E in equip&omacr;dio con I, e perci&ograve; $e di nuo|-
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347 uo la ragione che h&agrave;, il marco al re$tante H habbia la parte de $tate
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348 ra A C, a C M, $o$pe$o il marco L da M, far&agrave; equipondio
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349 con H: &amp; il corpo F E facea equipondio con I: $tar&agrave; dunque ogni co$a
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350 in equilibrio. $i &egrave; dunque ritrouato il ponto M, onde $o$pe$o il marco
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351 faccia equipondio con la grauezza data. Il che $i hauea da ritrouare.</I>
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352 <HEAD><I>PROPOSITIONE.</I>
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353 XII.</HEAD>
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354 <p>Fatta alla linea della $tatera application di corpo, e
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355 $o$pe$e in e$$a pi&ugrave; grauezze che $o$tentino un pe$o, ri|-
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356 trouare cia$cuna grauezza quanto portion di pe$o $o-
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357 $tenti.
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358 <HEAD><I>Dimo$tratione.</I></HEAD>
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359 <p><I>Sia la linea della $tatera A B il ponto del $o$tentimento C &amp; alla</I>
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360 <p n=>16</p>
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361 <FIG>
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362 <I>linea A B $ia fatta application di corpo et in e$$a appe$e le grauezze D
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363 E, F: D in A, E in G, F in H: e dette grauezze $o$tentine il pe$o I
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364 K L M: il cui momento $ia nel ponto B: $i cerca cia$cuna di dette gra-
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365 uezze D, E, F, quanta portione di pe$o $o$tenti. faccia$i per que$to alla
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366 linea B C uguale la C N: e la re$tante N A $i diuida in parti uguali nel
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367 p&omacr;to O, e quella ragione che h&agrave; B C a C O quell'habbia il corpo della $ta|-
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368 tera applicato ad N A ad M: $ar&agrave; d&utilde;que equip&otilde;der&atilde;te c&otilde; M: c la par
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369 te applicata ad N C &egrave; equiponder&atilde;te alla applicata &agrave; B C: d&utilde;que il cor-
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370 po della $tatera $t&agrave; in equip&otilde;dio con la portione del pe$o M: e le ragioni
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371 delle grauezze D, E, F, e delle di$tanze A C, G C, H C, cio&egrave; la ragio-
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372 ne della grauezza Dad F con la ragione della di$tanza A C a G C,
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373 compongon la ragion di P a Q. &amp; la ragione della grauezza E ad F,
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374 con la ragione della di$tanza G C ad H C, compongon la ragione</I>
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375 <p n=>17</p>
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376 <I>di Q ad R, &amp; in quella ragione che $ono le tre quantit&agrave;, P Q R,
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377 po$te in continua habitudine, nella i$te$$a $i di$tribui$ca il pe$o I K L:
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378 &egrave; manife$to per quel che $i &egrave; visto, che, D fa equiponderanza con I, lo
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379 E co'l K, e lo F con lo L: ilche $i cercaua.</I>
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380 <HEAD><I>PROPOSITIONE.</I>
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381 XIII.</HEAD>
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382 <p>La $tatera di grauezze appe$e, che facciano equipon|-
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383 dio: quantunque dal $ito orizontale mo$$a $i $t&agrave;.
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384 <FIG>
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385 <HEAD><I>Dimostratione.</I></HEAD>
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386 <p><I>Sia la $tatera nel $ito orizontale A B, il ponto della $o$pen$ione C,
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387 li pe$ie $ue centri D &amp; E, il centro commune di ambe le grauezze F;
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388 e mo$$a la statera del $ito orizontale, pa$$i il ponto A in G, il B in H,</I>
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389 <foot><I>C</I></foot>
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390 <p n=>18</p>
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391 <I>$i che habbia la $tatera la po$itione di G C H: li pe$i e $ui centri di, I e
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392 K: dico, che la $tatera G H $tar&agrave;, e non $i mouer&agrave; di $ito. $i mo$tra
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393 percioche e$$endo la grauezza I appe$a, inalzata, il centro $uo gi-
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394 rando verr&agrave; nella perpendicolare del ponto della $o$pen$ione: e perci&ograve;
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395 I, verr&agrave; nella perpendicolare del ponto G e K del ponto H. $ono
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396 dunque, G I H K parallele. e perche il centro commune de pe$i, diui-
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397 de nell'i$te$$a ragione la I K, &amp; la D E, e$$endo la ragione delli pe$i
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398 vn'i$te$$a, &amp; la C F nell'vna, e nell'altra $o$pen$ione perpendicolare,
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399 e parallela, co$i alle A D E B, come alle G I, K H. perci&ograve; diuidendo
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400 C F perpendicolare $imilmente la D E, &amp; la I K: $ar&agrave; il ponto F luo-
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401 go del centro nell'vna, luogo anco di centro nell'altra. e$$endo dunque
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402 il centro del pe$o commune nella perpendicolare della $o$pen$ione, $ta-
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403 r&agrave;. Ilche $i hauea da mo$trare.</I>
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404 <HEAD><I>PROPOSITIONE.</I>
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405 XV.</HEAD>
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406 <p>La $tatera di grauezze attaccate, che facciano equi-
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407 pondio, $e'l ponto della $o$pen$ione, non $ia nella linea
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408 delli centri: mo$$a dal $ito orizontale non $tar&agrave;, ma ri-
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409 tornar&agrave; nell'i$te$$o.
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410 <FIG>
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411 <HEAD><I>Dimo$tratione.</I></HEAD>
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412 <p><I>Sia la linea della $tatera, che $tia nel $ito horizontale A B, li pe-
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413 $i attaccati, &amp; li lor centri C e D, e diuida$i G D $econdo li pe$ireci-</I>
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414 <p n=>19</p>
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415 <I>pro camente nel ponto E: &egrave; manife$to che'l ponto E $ia il centro com-
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416 mune di ambi li pe$i, e che mentre la $tatera $ta, che $ia detto centro
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417 nella perpendicolare, che cala dal ponto F. perche dunque li pe$i $ono
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418 alla statera affi$$i, e non mutano li centri po$itura con la linea A B, e
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419 $empre fanno con e$$a angoli retti le C A, D B, E F, perci&ograve; mo$$a
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420 la $tatera dal $ito horizontale, non $ar&agrave; E centro c&omacr;mune nella perpen-
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421 dicolare della $o$pen$ione: ma girando v$cir&agrave; di detta perpendicolare,
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422 e perci&ograve; la $tatera non $tar&agrave;, $in che di nuouo il detto ponto non torne
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423 nella perpendicolare.</I>
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424 <FIG>
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425 <foot>C 2</foot>
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426 <p n=>20</p>
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427 <HEAD>VETTE, E
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428 LEVA.</HEAD>
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429 <HEAD><I>DEFINITION.</I>
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430 I.</HEAD>
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431 <p>Vette diciamo la linea, che $o$tiene grauezza,
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432 qual $ia nelli $ue ponti e$tremi $o$tenuta.
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433 <HEAD><I>DEFINITION.</I>
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434 II.</HEAD>
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435 <p>Et altrimente, vette motiua e leua, la linea che $o-
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436 $tenga grauezza, $tabilita in vn ponto che $otto leua
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437 diciamo, &amp; in vn'altro ponto da po$$anza, o mo$$a, o
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438 $o$tenuta.
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439 <HEAD><I>POSITION.</I>
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440 I.</HEAD>
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441 <p>Mi$uriamo la po$$anza con vna grauezza equiualen|-
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442 te, o appe$a nell'i$te$$o ponto della po$$anza, o nell'al-
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443 tro ponto egualmente dal $ottoleua di$co$to.
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444 <HEAD><I>POSITION.</I>
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445 II.</HEAD>
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446 <p>Cia$cuna po$$anza in quanto $o$tiene, e$$ere egua-
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447 le al pe$o $o$tenuto.
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448 <FIG>
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449 <p n=>21</p>
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450 <HEAD><I>PROPOSITION.</I>
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451 I.</HEAD>
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452 <p>S'il $ottoleua $tia tra la grauezza, e la po$$anza che
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453 $o$tenga detta grauezza; $ar&agrave; tra la po$$anza, &amp; il pe-
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454 $o la ragione, che &egrave; tra le parti della leua, reciprocamen|-
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455 te.
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456 <FIG>
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457 <HEAD><I>Dimo$tratione.</I></HEAD>
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458 <p><I>Sia la linea A B, il ponto del $ottoleua C, la grauezza D $oste-
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459 nuta nel ponto della leua A; la po$$anza che $o$tenga detta graue z-
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460 za in B: dico che la po$$anza B al pe$o D, ha quella ragione che ha
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461 la parte di leua A C alla C B, qual &egrave; ragion reciproca. $i mo$tra: inten|-
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462 da$i attaccato in B il pe$o che faccia equipondio con D: &egrave; manife$to che
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463 detto pe$o E $ia equiualente alla forza B, ma il pe$o E al pe$o Dhala
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464 ragione che A C a C B, che &egrave; la ragione reciproca di grauezza, e di-
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465 $tanze: dunque, la potenza ancora haue l'i$te$$a ragione. ilche$i ha-
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466 uea da mo$trare.</I>
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467 <HEAD><I>PROPOSITION.</I>
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468 II.</HEAD>
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469 <p>Se due potenze $o$tentino vna grauezza con vn vet-
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470 te, cia$cuna $o$tentar&agrave; la $ua portione, $econdo l'inter-
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471 uallo del pe$o dalle potenze, pigliato reciprocamente.
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472 <p n=>22</p>
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473 <FIG>
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474 <HEAD><I>Dimo$tratione.</I></HEAD>
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475 <p><I>Sia il vette A B, dal cui ponto D, penda il pe$o C: le potenze che
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476 $o$tengono dette grauezze $iano A &amp; B: dico che'l B, e lo A $o$ten-
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477 tano portioni proportionali all'interualli reciprocamente: cio &egrave; che
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478 quella ragione c'ha l'interuallo, B D, a D A, quella h&agrave;bbia la por-
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479 tione $o$tentata dall' A, alla portione $o$tentata dal B, $i dimo$tra:
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480 tagli$i ad A D uguale B E, accoppiata dunque communemente la D
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481 E, $ar&agrave; A E uguale a B D: ag giunga$i all' A e la A G, che le $ia egua-
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482 le, &amp; ad E B la B F che $imilmente le $ia eguale. $ar&agrave; di tutta la G F,
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483 il ponto mezzano D, &amp; della G E, il ponto mezzano A, &amp; della E
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484 F, il ponto mezzano B. applicata dunque a tutta la G F, una grauez-
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485 za che $ia uguale a C, $ar&agrave; di detta grauezza il ponto di momento in D
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486 &amp; $ar&agrave; equiualente nella $ua operatione alla grauezza C, &amp; di e$$a
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487 la parte applicata a G E ha il $uo momento in A, c la parte applica-
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488 ta ad E ha il $uo momento in B. dunque della grauezza applicata
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489 la potenza A, ne $o$tentar&agrave; la portione applicata a G E: e la potenza
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490 B, la portione applicata ad E F. Ma G E ad E F, ha la ragione che
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491 l'interuallo B D, a D A che &egrave; reciproca. dunque le potenze $o$tenta-
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492 no le portioni de'pe$i proportionali, reciprocamente pigliate con l'inter
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493 ualli. ilche $i hauea da mo$trare.</I>
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494 <p n=>23</p>
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495 <HEAD><I>PROPOSITIONE.</I>
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496 III.</HEAD>
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497 <p>Se il $ottoleua $ia fuori della grauezza, e della po$-
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498 $anza, $ar&agrave; la ragion della po$$anza alla grauezza l'i$te$
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499 $a, che dell'interualli da e$se al $ottoleua reciprocam&etilde;-
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500 te pigliati
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501 <FIG>
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502 <HEAD><I>Dimo$tratione.</I></HEAD>
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503 <p><I>Sia la leua A B il $otto lcua A, la grauezza C, il $uo momento in
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504 D, la po$$anza che $o$tiene in B: dico che la po$$anza alla grauezza
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505 ha la ragione, che D A ad A B, che &egrave; la ragion delle di$tanze piglia-
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506 te dal $ottoleua reciprocamente: $i mo$tra: perche il pe$o C, e $o$ten-
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507 tato dalla leua B A, e la leua &egrave; $o$tentata in due ponti B &amp; A. dunque
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508 il pe$o &egrave; $o$tentato dalle potenze in B &amp; A compartitamente, cio<*>
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509 la po$$anza B $o$tenta tal portion di pe$o, qual'&egrave; la di$tanza A D di A
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510 B, &amp; A, tal portione qual'&egrave; D B, di B A, e perche la po$$anza $o-
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511 $tenente &egrave; uguale al pe$o che $o$tiene, $ono ambe le po$$anze B &amp; A
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512 giuntamente pigliate uguali al pe$o E; e la portione $o$tentata da B:
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513 al tutto harr&agrave; quella ragione che la portion della leua D A a tutta
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514 la leua A B. qual &egrave; l'i$te$$a che della di$tanza della grauezza, alla di-
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515 $tanza della potenza. $i ha dunque l'intento.</I>
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516 <p n=>24</p>
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517 <HEAD><I>PROPOSITIONE.</I>
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518 IV.</HEAD>
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519 <p>Se vna grauezza $ia con vna leua $o$tenuta da due
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520 ponti; &amp; accrefciuta la leua dall altra parte $i appenda
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521 grauezza equiponderante, &amp; $i tra $muti in $tatera: $o-
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522 itentar&agrave; il $o$tenimento in tal commutatione pe$o mag
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523 giore, quale al pe$o di prima $o$tenuto, ha ragione com
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524 po$ta della ragione delle portioni di tutta la linea accre
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525 $ciuta communicanti, alle portioni interuallate: fat-
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526 te le due diui$ioni al ponto del $ottoleua, &amp; al ponto
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527 del primo momento.
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528 <FIG>
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529 <HEAD><I>Dimo$tratione.</I></HEAD>
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530 <p><I>Sia la leua A B, il $ottoleua in A: lagrauezza $o$tenuta in C, la
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531 po$$anza che'l $o$tiene in B. &amp; allungata la B A in vn D, appenda-
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532 $i in D, vna grauezza che $o$tenti la grauezza C. dico che in que$ta
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533 commutatione il $ottoleua A $o$tenti pe$o maggiore, &amp; che ilpe$o
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534 $o$tenuto in detta commutatione, alpe$o $o$tenuto di prima, ha la ra&shy;
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535 gion compo$ta delle D C, A D, parti communicanti, alle D A, a C
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536 B, parti interuallate. $i mo$tra: perche la parte del pe$o $o$tenuto da
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537 A, a tutto il pe$o C, ha la ragione, che B C a B A: &amp;. il pe$o C, ad
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538 ambi li pe$i C &amp; D, ha la ragione che D A a D C, ma la ragione del-</I>
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539 <p n=>25</p>
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540 <I>la portione $o$tenuta da A, alla grauezza C, &amp; di C, ad ambe CD, $ot
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541 trattone il termine mezzano, compongono la ragione della portione $o$te
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542 nuta da A, ad ambe le C D, &amp; la ragione di B C a BA, &amp; di D A a D
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543 C, fanno la ragione compo$ta delle parti communicanti alle interuallate.
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544 Ha$$i dunque l'intento: che'l pe$o di prima $o$tenuto, al pe$o $o$tenuto
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545 dopo la commutatione, ha la ragion compo$ta delle parti interuallate alle
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546 communicanti. Ilche $i hauea da mo$trare.</I>
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547 <HEAD><I>PROPOSITIONE.</I>
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548 V.</HEAD>
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549 <p>Date nell' e$tremit&agrave; del vette due po$sanze c'habbia-
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550 no qual$iuoglia ragione tra di loro; e dato vn pe$o, a det-
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551 te po$$anze giuntamente pigliate vguale, ritrouare il
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552 ponto del vette, onde il dato pe$o $o$pe$o, $ia da det
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553 te po$$anze $o$tenuto.
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554 <FIG>
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555 <HEAD><I>Dimo$tratione.</I></HEAD>
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556 <p><I>Sia il vette AB: le po$$anze nelli ponti A e B, c'habbiano tradi lo-
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557 ro qual$iuoglia ragione: &amp; il pe$o ad ambe po$$anze giuntamente pi-
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558 gliate vguale $ia C: $i cerca il ponto, onde detto pe$o $ia da dette po<32>&atilde;
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559 ze $o$tenuto. per il che dico: che $e in quella ragione, c' ha la po$$an-</I>
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560 <foot><I>D</I></foot>
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561 <p n=>26</p>
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562 <I>za B, $i diuida la vette AB in D, e $ia come la po$$anza Aalla B: cos&igrave;,
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563 la portione di vette B D a D A: dico che po$to il pe$o C, in D: $ar&agrave;,
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564 $o$tenuto da dette po$$anze: percioche grauando il pe$o nelli ponti B: &amp;
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565 A, cbe $o$tentano compartitamente, $econdo la ragion di BD a DA:
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566 &amp; hauendo la portion che graua in A, alla portion che graua in B, la
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567 ragion che B D a D A: qual'&egrave; l'i$te$$a che della po$$anza A alla po$$anza
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568 B: dunqne la portione che graua in, A alla portione che graua in B,
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569 e come la po$$anza A, alla B: e permut&atilde;do la portion che graua in A, a
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570 la po$$anza A, $ar&agrave; come la portione che graua in B alla po$$anza B,
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571 e componendo li antecedenti, tutto il pe$o C, ad ambe le po$$anze giun
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572 te, harr&agrave; l'i$te$$a ragione che vna advna: ma il pe$o tutto C, &egrave; vgua-
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573 le ad ambe le po$$anze giuntamente pigliate: dunque diui$amente le
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574 portioni, cia$cuna alla po$$anza oue graua, $ar&agrave; vguale: e percio $a-
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575 r&agrave; del pe$o $o$tenuto, la portione che graua in A, vguale alla po$$anza
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576 in A: e la portione che graua in B, vguale alla p&ograve;$$anza iu B: e percio
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577 le po$$anze $o$tentaranno il detto pe$o nel ponto D. Il che $i hauea da
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578 mo$trare,</I>
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579 <HEAD><I>Appendice.</I></HEAD>
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580 <p>Et &egrave; manife$to che in ogni altro ponto del detto vet
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581 te, il pe$o non $ar&agrave; $o$tenuto, ma aggrauer&agrave; pi&ugrave; l'vna &ograve;
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582 l'aitra po$$anza, ver$o oue $ar&agrave; portato.
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583 <HEAD><I>PROPOSITIONE.</I>
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584 VI.</HEAD>
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585 <p>Se una leua $ia inalzata, oba$$ata $otto l'orizonte:
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586 &amp; da un ponto fuori di e$$a, $i tireranno due perpendi-
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587 colari, l'vna ad e$$a leua, e l'altra all'orizonte: faran
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588 no le due perpendicolari angolo tra di loro, vguale all'
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589 angolo della leua con l'orizonte.
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590 <p n=>27</p>
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591 <FIG>
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592 <HEAD><I>Dimo$tratione.</I></HEAD>
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593 <p><I>Sia la linea orizontale A B, la leua $opra di e$$a inalzata o de$pre$-
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594 $a A C. il ponto fuori della leua E: da cui $i tirino due perpendicolari
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595 l'vna alla leua DE, l'altra all'orizonte D F, che $eghi la leua in F, &amp;
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596 la linea orizontale in G. dico che l'angolo fatto dalle due D E, D F
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597 $ia vguale allangolo fatto, dalle due A B, A C: $i mo$tra: percioche
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598 le due A C, D G, $i $egano nel ponto F, $aranno l'angoli A F G, et D
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599 F E, d'incontr vguali: e gli angoli ad E &amp; G $ono retti: dunque il tri-
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600 angolo D F E, &egrave; equiangolo al triangolo A F G, e l'angolo F D E, v-
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601 guale a l'angolo F A G. Il che $i hauea da m&ograve;$trare.</I>
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602 <HEAD><I>Appendice,</I></HEAD>
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603 <p>Et &egrave; manife$to che e$$endo detto ponto di $opra la li
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604 nea della leua inalzata, e di $otto della leua ba$$ata; $e-
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605 cher&agrave; detta linea in ponto pi&ugrave; dalla po$$anza lontano.
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606 e per c&otilde;trario pigliando$i detto ponto, o $otto dell'alza-
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607 t&agrave;, o$opra della ba$$ata, $egher&agrave; in ponti pi&ugrave; &agrave; detta pos-
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608 $anza vicini.
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609 <foot>D 2</foot>
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610 <p n=>28</p>
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611 <HEAD><I>PROPOSITIONE.</I>
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612 VII.</HEAD>
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613 <p>Se'l centro del pe$o attaccato ad e$$a leua $ia $opra
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614 della leua, inalzata la leua, la po$$anza $o$tentar&agrave; minor
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615 pe$o.
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616 <FIG>
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617 <HEAD><I>Dimo$tratione.</I></HEAD>
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618 <p><I>Sia la leua A B, a cui $ia attaccata vna grauezza, il cui centro $ia
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619 C: &amp; intenda$i detta leua in $ito dall'orizonte eleuato: dico che la po-
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620 tenza B, $o$tenta del pe$o della grauezza minor portione, che nel $ito
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621 orizontale. $i mo$tra: tirin$i dal ponto C linee, l'vna perpendicolare
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622 alla leua che $ia C D, &amp; l'altra perpendicolare all'orizonte, che
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623 $ia C E, che $eghi la leua nel ponto E: &egrave; manife$to che'l detto ponto
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624 $ar&agrave; pi&ugrave; di$co$to dalla po$$anza, e pi&ugrave; vicino al ponto del $ottoleua. $e
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625 dunque per lo ponto C, $i tiri la linea G C F, parallela all'orizonte, &amp;
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626 per li ponti B &amp; A, lelinee B F, A G, perpendicolari all'oriz&otilde;te &egrave; mani
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627 fe$to, che l'i$te$$o effetto fa la po$$anza in F che $e fu$$e in B, e lo $o$tegno
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628 in A l'i$teffo che $e fu$$e in C: percioche cia$cun momento opera $econ
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629 da la $ua perpendicolare: perche dunque po$ta la po$$anza in F, e lo $o-</I>
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630 <p n=>29</p>
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631 <I>$tegno in G, la po$$anza F, $o$tiene tal portione di tutto ilpe$o, qual
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632 portione &egrave; G C, di G F: e qual'&egrave; G C, di tutta G F, tal'&egrave; A E di tutta
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633 A B, perche le A G, C E, B F, $ono parallele: $o$tenta dunque la po$-
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634 $anza B, del pe$o tal portione, qual'&egrave; A E ditutta A B: $e dunque
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635 A E &egrave; minor portione di A B, che la A D, dell'i$te$$a A B: la po$$an
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636 za con la leua inalzata il cui centro del pe$o &egrave; $opra, $o$tenta minor
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637 portione che nel $ito orizontale. Il che $i hauea damo$trare.</I>
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638 <HEAD><I>Appendice. I.</I></HEAD>
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639 <p>E per l'i$te$$o mezzo $i mo$trer&agrave; che quanto pi&ugrave; la le
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640 ua s'inalza, tanto minor pe$o $o$tiene.
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641 <HEAD><I>Appendice. II.</I></HEAD>
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642 <p>E che po$to il centro della grauezza $otto la leua,
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643 quanto pi&ugrave; s'inalzi, magior portione di pe$o $o$tenga.
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644 <HEAD><I>Appendice. III.</I></HEAD>
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645 <p>E che nelle leue ba$$ate $otto l'orizonte, auuenga a
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646 contrario.
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647 <HEAD><I>PROPOSITIONE.</I>
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648 VIII.</HEAD>
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649 <p>Dato nella leua il ponto di momento di una grauez-
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650 za, e data qual$ivoglia ragione di po$$anza a grauez-
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651 za, ritrouar nella leua il ponto, oue la data po$$anza $o
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652 $tenga la data grauezza.
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653 <p n=>30</p>
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654 <FIG>
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655 <HEAD><I>Dimo$tratione.</I></HEAD>
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656 <p><I>Sia nella leua A B, il ponto del $ottoleua in A: il ponto di momen-
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657 to della data grauezza in C. et la ragion della po$$anza data alla grauez
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658 za, come di E a D: $i cerca nella leua il ponto, oue po$ta la data po$-
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659 $anza $o$tenga la data grauezza. per que$to: faccia$i come E a D, cos&igrave;
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660 A C ad A F: &amp; intenda$i la po$$anza in F. dico che detta po$$anza in
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661 F $o$tiene la grauezza in C. $i mo$tra: percioche e$$endo la ragion del-
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662 la po$$anza alla grauezza come E a D, e la ragion dell'interuallo del
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663 la grauezza A C, all'interuallo della po$$anza A F, l'i$te$$a reciproca
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664 mente: $o$tentar&agrave; dunque la data po$$anza in F, la grauezza in C. Il
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665 che $i cercaua.</I>
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666 <HEAD><I>Appendice.</I></HEAD>
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667 <p>Et &egrave; manife$to che in qual $i uoglia altro ponto oltre
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668 del termine del $o$tenimento, la data po$$anza mouer&agrave;
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669 la data grauezza: e tanto pi&ugrave; facilmente quanto pi&ugrave; $i
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670 $co$tar&agrave;.
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671 <FIG>
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672 <p n=>31</p>
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673 <HEAD>RAGGINELL
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674 ASSE.</HEAD>
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675 <HEAD><I>SVPPOSITIONE.</I></HEAD>
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676 <p>Svpponiamo, in vno i$te$$o a$$e, due rag
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677 gic'habbiano nelli $uoi $tremi li centri de pe$i.
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678 <p>E detti raggi, o in vna pianezza, e che non facciano
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679 angolo, o in due, e che facciano angolo.
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680 <HEAD><I>POSITIONE.</I></HEAD>
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681 <p>Pigliamo, il momento di cia$cun pe$o, $econdo il pon
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682 to, oue la perpendicolare del momento taglia la linea
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683 orizontale, che pa$$a per l'a$$e.
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684 <FIG>
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685 <HEAD><I>PROPOSITIONE.</I>
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686 I.</HEAD>
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687 <p>Delle grauezze po$te in raggi che non fanno tra di
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688 loro angolo, in qualunque $ito po$te, li momenti tra di
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689 loro hanno l'i$te$$a ragione.
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690 <HEAD><I>Dimo$tratione.</I></HEAD>
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691 <p><I>Sial'a$$e A, a cui $iano affi$$i li raggi A B, A C, qualis'intenda-</I>
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692 <p n=>32</p>
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693 <FIG>
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694 <I>no e$$ere nel$ito orizontale, &amp; moua$i dal detto $ito, s&igrave; che il B uen
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695 gain D, &amp; il C venga in E: dico che li momenti delle grauezze in det
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696 ti raggi quantunque mo$$i di $ito, $iano nell'i$te$$a ragione tra di loro.
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697 $i mo$tra: tiri$iper D la perpendicolare D F &amp; per E la perpendicola
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698 re E G; perche dunque F A ad A G, ha la ragione che D A ad A E,
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699 perci&ograve; che $ono D F, E G, parallele: ma come D A ad A E, cos&igrave; B A ad
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700 A C: perche $ono l'i$te$$i raggi, come dunque B A ad A C, cos&igrave; F A
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701 ad A G: e perche la ragion delli momentie compo$ta della ragion delle
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702 grauezze, e della ragion delle di$tanze dal centro: ma la ragione delle
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703 grauezze &egrave; l'i$te$$a: e la ragione delle di$t&atilde;ze &egrave; l'i$te$$a: dunque la ragion
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704 di ambe compo$te, &egrave; anco l'i$te$$a. Il che $i hauea da mo$trare.</I>
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705 <p n=>33</p>
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706 <HEAD><I>PROPOSITIONE.</I>
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707 II.</HEAD>
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708 <p>Date qual $i uoglia due grauezze, nelli raggiche fac-
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709 ciano angolo dato, ritrouar nelle loro circolationi, pon-
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710 ti oue facciano equipondio.
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711 <FIG>
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712 <HEAD><I>Dimo$tratione.</I></HEAD>
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713 <p><I>Sia l'a$$e A: li raggi che facciano &atilde;golo dato A C, B A: &amp; int&etilde;da$inel-
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714 li ponti B e C, e$$er li c&etilde;tri delle grauezze: &amp; le circonferenze che det-
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715 ti ponti girando attorno fanno, $iano E B, C F: $i cercano in dette cir-</I>
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716 <foot><I>E</I></foot>
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717 <p n=>34</p>
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718 <I>conferenze li ponti, oue e$$endo dette grauezze, facciano equipondio.
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719 Diuida$i la B C interuallo de centri, $iche qual ragione ha la grauczza,
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720 B, alla C, tal habbia la linea C D alla, D B: e tiri$i A D: e tirata
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721 per A, la A E B perpendicolare all'Orizonte, faccia$i all'angolo D A
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722 B, vguale lo E A G: &amp; allo D A C, vguale E A H: dico che'l ponto
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723 G, &egrave; oue portato il B, &amp; H, oue portato il C, fanno equipondio. E prima
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724 che portato il B in G, venga il C in H, &egrave; manife$to: percioche l'ango
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725 B A C &egrave; vguale al G A H: e perl'i$te$$a ragione, &egrave; manife$to che nell'
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726 i$te$$o tempo il ponto D, $ia nella A E. ma il p&otilde;to D &egrave; il centro commu-
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727 ne di pe$o di dette due grauezze. E dunque il centro commune nel
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728 la perpendicolare del $o$tenimento: e perci&ograve; le grauezze $tanno. Jl che
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729 $i cercaua.</I>
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730 <HEAD><I>Appendice. I.</I></HEAD>
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731 <p>Et &egrave; manife$to che nelli due ponti, oppo$ti alli ritroua
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732 ti, facciano equipondio: &amp; non altroue: percioche in o-
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733 gni altra po$itura oltre di dette due, il centro commune
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734 e fuori del perpendicolo.
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735 <HEAD><I>Appendice. II.</I></HEAD>
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736 <p>Et &egrave; manife$to che nell'arco $otto il ponto dell'equi
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737 pondio la grauezza ha momento maggiore: e nell'arco
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738 $oprail ponto dell'equipondio ha momento minore.
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739 <HEAD><I>PROPOSITIONE.</I>
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740 I.</HEAD>
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741 <p>D&agrave;te qual $i uoglia due grauezze nelli dati raggi, che
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742 fanno dato angolo: ritrouar nelle loro circolationi, pon-
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743 <p n=>35</p>
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744 ti oue ilmomento dell'uno, al mom&etilde;to dell'altro habbia
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745 qual $i voglia data ragione.
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746 <FIG>
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747 <HEAD><I>Dimo$tratione.</I></HEAD>
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748 <p><I>Siano le date grauezze A &amp; B: liraggi AC, BC, fi$$i nell'a$$e C:
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749 che facciano dato &atilde;golo: e la circolation di A $ia, AD: di B $ia B E.e
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750 la data qual$iuoglia ragione $ia diFaG: $icercano nella circolatione DA
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751 e nella BE, ponti oue habbian li momenti di A e B, ragion di F a G. In|-
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752 tenda$i nella ragion di A a B, la quantit&agrave; F ad H. e nella re$tante ra-
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753 gione di H a G, $i diuida A B in L. &amp; all'angolo L C A faccia$i vgua-</I>
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754 <foot><I>D</I> 2</foot>
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755 <p n=>36</p>
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756 <I>le il D C M, &amp; allo L C B eguale il DCN: &egrave;manife$to, che portato A in
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757 M: B verr&agrave; in N. &amp; il ponto L nella perpendicolare C D. e$e per il
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758 ponto C. $i tiri la P C Q parallela all'Orizonte: e dalli ponti M &amp; N $i
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759 tirino a que$ta, perpendicolari le MQ NP: $ar&agrave; il momento della
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760 grauezza in M, al momento della grauezza in N di ragion compo
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761 $ta della grauezza A alla grauezza B, e della distanza Q C, alla CP,
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762 che &egrave; l'i$te$$a che di A L ad L B.percioche que$ta &egrave; l'i$te$$a che di M O ad
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763 O N: cio&egrave; della compo$ta delle ragioni di F ad H, e di H a G: ci&ograve; &egrave; di F a
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764 G. harr&atilde;no dunque li mom&etilde;ti di A &amp; B, mentre $iano po$ti nelli ponti
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765 M &amp; N la ragiondata di F a G. Il che $i cercaua.</I>
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766 <HEAD><I>Appendice.</I></HEAD>
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767 <p>Et &egrave; manife$to che prodotte le linee del centro nelli
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768 ponti oppo$ti delle dette circonferenze, hauranno iui li
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769 momenti delle date grauezze l'i$te$$a ragione: enon
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770 altroue.
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771 <FIG>
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772 <p n=>37</p>
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773 <HEAD>MOMENTI
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774 CENTRALI</HEAD>
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775 <p>Eqvanto delli momenti paralleli habbiamo
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776 mo$trato, tutto $i adatter&agrave; anco alli momenti con-
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777 correnti &agrave; centro: $e in vece di linee dritte con$ideria-
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778 mo le circolari d'intorno il centro oue li momenti con-
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779 corrono: &amp; in dettecircolari $i faccia l'i$te$$a partitione:
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780 e $e in vece delli corpi terminati, da $uperficie parallele,
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781 s'intendano altri corpi terminati, parte da $uperficie sfe
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782 riche c'habbiano detto centro: parte da $uperficie pia
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783 ne che pa$$ino per e$$o.
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784 <FIG>
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785 <p n=>38</p>
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786 <HEAD>ROTE VET-
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787 TIVE.</HEAD>
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788 <HEAD><I>SVPPOSITIONE.</I></HEAD>
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789 <p>Svpponiamo vna, o pi&ugrave; rote congiogate,
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790 muouer$i per piano, che $ia, o di po$itura orizontale,
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791 o inchinata.
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792 <HEAD><I>DEFINITION.</I>
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793 I.</HEAD>
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794 <p>Cogiogation $emplice, diciamo dellerote, che $ono
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795 s&ugrave; di vn'i$te$$o a$$e.
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796 <HEAD>I.</HEAD>
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797 <p>Molteplice, delle rote che $ono in pi&ugrave; a$$i.
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798 <HEAD>III.</HEAD>
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799 <p>Portioni terminate dal $o$tenimento diciamo nel cir
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800 colo, le fatte dalla linea perpendicolare per lo ponto del
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801 contatto, all'orizonte: e nel cilindro, dalla $uperficie pia
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802 na per la linea del contatto, perpendicolare $imilmente
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803 all'orizonte.
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804 <p n=>39</p>
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805 <HEAD><I>POSITIONE.</I>
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806 I.</HEAD>
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807 <p>Poniamo ogni forza, o trattiua, o pul$iua, giunger mo
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808 mento uer$o quella parte, oue tira, o $pinge.
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809 <HEAD>II.</HEAD>
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810 <p>E $e'l centro del pe$o $ia nell'i$te$$a linea dell'appendi
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811 mento, o $o$tenimento: che la grauezza non habbia mo
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812 mento, ne uer$o l'vna, ne uer$o l'altra parte.
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813 <FIG>
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814 <HEAD><I>PROPOSITIONE.</I>
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815 I.</HEAD>
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816 <p>Dellarota vettiua, che $i moue $opra di vn piano ori-
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817 zontale, il centro del pe$o $empre &egrave; nella perpendicola-
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818 re del $o$tenimento.
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819 <FIG>
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820 <p n=>40</p>
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821 <HEAD><I>Dimo$tratione.</I></HEAD>
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822 <p><I>Sia la linea Orrzontale A B: il circolo che rappre $enta la rota, CD:
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823 il ponto, oue detta rota tocca il piano C: da cui $i cacci ad angoli ret-
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824 ti la linea C D, &egrave; manife$to che detta linea, &egrave; la perpendicolare del $o$te
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825 nimento: &amp; perquelche nelli libri Giometrici $i mo$tra: che pa$$a per il
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826 centro del circolo, che &egrave; il centro della rota e grauezza: perilche diui-
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827 de il circolo il parti vguali, &amp; equeponderanti: &egrave; dunque il centro
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828 del pe$o nella perpendicolare del $o$tenimento. Il che $i hauea da
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829 mo$trare,</I>
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830 <HEAD><I>Appendice. I.</I></HEAD>
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831 <p>Et il $imile $i mo$tra, nelle $emplici rote congiogate,
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832 $opra l'a$$e de quali, po$i la grauezza.
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833 <HEAD><I>Appendice, II.</I></HEAD>
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834 <p>Et &egrave; manife$to nelle rote, s&ugrave; l'a$$e de quali po$i la
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835 grauezza: che nel piano oriz&otilde; tale, non habbian momen&shy;
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836 to ne ver$o l'vna, ne ver$o l'altra parte.
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837 <HEAD><I>Appendice. III.</I></HEAD>
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838 <p>E che perci&ograve; qual $i voglia po$$anza, le porter&agrave; cos&igrave;
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839 nell'vna, come nell'altra parte,
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840 <p n=>41</p>
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841 <HEAD><I>PROPOSITIONE.</I>
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842 II.</HEAD>
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843 <p>Nella rota che $i porta per piano inchinato, il centro
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844 del pe$o, &egrave; fuori della perpendicolare del $o$tenimento.
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845 et il momento della rota appoggiata al piano, al momen&shy;
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846 to della rota $o$pe$a, la ha ragione, che l'ecce$$o delle
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847 portioni del circolo, al circolo tutto.
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848 <FIG>
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849 <foot>F</foot>
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850 <p n=>42</p>
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851 <HEAD><I>Dimo$tratione.</I></HEAD>
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852 <p><I>Sia la linea che rappre $enta il piano orizontale A B: la linea del pia
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853 no inchinato A C: il circolo della rota D E FG: il toccamento D: e dal
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854 ponto D, tiri$i perpendicolare all'orizonte B D F: &egrave; manife$to che detta
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855 linea, $ia la perpendicolare del $o$tenimento: dico che'l centro del pe$o
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856 &egrave; fuori di detta linea. Si mo$tra: perche del triangolo D B A: l'angolo,
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857 E, che fa la perpendicolare con l'orizonte, &egrave; retto: re$ta l'angolo B D A,
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858 a cuto: e perci&ograve; la portione D G F, e maggiore del $emicircolo; &amp; in e$
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859 $a $ar&agrave; il centro del circolo, che &egrave; anco centro di pe$o. &egrave; dunque il cen-
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860 tro del pe$o fuori della linea del $o$tenim&etilde;to. De $criua$i alla D E, la por
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861 tione di cir colo D H F, $imile a D E F; $aranno dette portioni vgua-
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862 li, e faranno equipondio. re$ta dunque la figura lunare $enza equi
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863 pondio: &amp; il momento della rota appoggiata $ar&agrave; meno che della ro
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864 ta $o$pe$a, $econdo la ragione della figura lunare a tutto il circolo: cio &egrave;
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865 $econdo la ragione dell'ecce$$o delle portioni, al circolo tutto. Il che
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866 $i hauea da mo$trare.</I>
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867 <HEAD><I>Appendice. I.</I></HEAD>
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868 <p>E l'i$te$$o che si &egrave; mo$trato nella rota c'ha grauezza;
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869 si mo$tra nelle rote al cui a$$e appoggi altro pe$o.
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870 <p><I>Percio che $e in vece delpe$o appoggiato all'a$$e, intendiamo dar$i
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871 l'i$te$$o pe$o alle rote: e$$endo pe$i vguali con loro centri nell'i$te$$e li-
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872 nee, &amp; la linea del $o$tenimento l'i$te$$a, harranno li pe$i l'i$te$$i mom&etilde;ti</I>
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873 <HEAD><I>Appendice, II.</I></HEAD>
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874 <p>Et &egrave; manife$to che detta rota correr&agrave; ver$o la parte
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875 del piano inferiore.
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876 <p><I>Percioche tirata dal centro I, la IG K perpendicolare del momento
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877 tutto $in che s'inc&otilde;tri col piano per oue camina: $ar&agrave; il ponto G della cir</I>
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878 <p n=>43</p>
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879 <I>conferenza di$co$to dal p&otilde;to K del piano per oue camina la rota: e t&atilde;to
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880 maggiorm&etilde;te il p&otilde;to oue s'inc&otilde;tra la perp&etilde;dicolare del c&etilde;tro di pe$o del
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881 la figura lunare: la cui di$t&atilde;za dalla linea del $o$tenim&etilde;to, &egrave; maggior che
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882 la di$tanza del centro del circolo, $econdo la ragion di tutto il circolo al
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883 la figura lunare.</I>
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884 <HEAD><I>PROPOSITIONE.</I>
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885 III.</HEAD>
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886 <p>Se vn pe$o $ia portato da due c&otilde;giogationi di rote,
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887 $ar&agrave; il pe$o $o$tenuto dalli due a$$i compartitamente, $e-
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888 condo la ragione delle di$tanze del momento da gli a$$i,
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889 reciprocamente.
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890 <FIG>
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891 <HEAD><I>Dimo$tratione.</I></HEAD>
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892 <p><I>Siano le due congiogationi di rote rappre$entate con li due circoli,
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893 de quali gli c&etilde;tri $ono A e B ponti, che rappre$entano li due a$$i: e dal
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894 ponto A al B, tiri$i la A B. &amp; intenda$i il centro del pe$o tutto appo
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895 giato a detti due a$$i hauere il momento nel ponto C della detta linea.</I>
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896 <foot><I>F</I> 2</foot>
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897 <p n=>44</p>
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898 <I>Dico che'ldetto pe$o &egrave; $o$tenuto da detti a$$i compartitamente, $econdo
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899 la ragione delle BC, AC: cio&egrave; che di tutto il pe$o l'a$$e A. ne $o$ten-
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900 ter&agrave; tal portione qual'&egrave; BC di B A, e B tale qual'&egrave; AC di AB, Si mo
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901 $tra intenda$i prol&otilde;gata la AB nell'vna e l'altra banda, far$i ad AC
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902 vguale la BD: &amp; alla BC, vguale la AE: $aranno le EC, DC vguali:
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903 e di nuouo fatto alla AC uguale la AE, $aranno le DB, BF, e le AE
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904 AF, vguali: epercio $e alla linea DE, s'intenda fatta application di
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905 corpo: il momento di tutto $ar&agrave; nel ponto C. di cui il detto a$$e A ne
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906 $o$tentar&agrave; la portione applicata ad EF: e l'a$$e B la portione applicata
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907 a DF, la ragion de quali &egrave; l'i$te$$a: che di BC ad AC: ma del corpo ap
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908 plicato il centro del pe$o &egrave; l'i$te$$o, dall'i$te$$i ponti $o$tenuto. $o$tengono
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909 dunque gli a$$i il pe$o compartitamente $econdo la ragion di BC a C-
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910 A. Il che $i hauea da mostrare.</I>
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911 <HEAD><I>PROPOSITIONE.</I>
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912 IIII.</HEAD>
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913 <p>Se'l pe$o sia portato da due congi&otilde;gationi di rote per
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914 piano inchinato: $o$t&etilde;ntar&agrave; l'a$$e dellerote inferiori di
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915 detto pe$o, ma ggior portione che $e fu$$e nel piano ori
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916 zontale.
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917 <HEAD><I>Dimo$tratione.</I></HEAD>
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918 <p><I>Sia la linea del piano orizontale AB: del piano inchinato AC: li
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919 centri de circoli delle rote D, &amp; E: il centro della grauezza che s&ugrave;
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920 gli a$$i di dette rote appoggia F: Dico che di detta grauezza, dall'a$$e
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921 D, ne $ar&agrave; $ostentata maggior portione: e dall'a$$e E, minore, che $e
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922 portata fu$$e per piano Orizontale. Si mo$tra: tiri$i da F perpendico
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923 lare alla DE, che $ia FG: e perpendicolare all'orizonte che $ia FH: $a
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924 r&agrave; il ponto G, il ponto del momento nel $ito orizontale. &amp; H, nell'in
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925 chinato: e perche EH, &egrave; maggior portione di ED: che EG, e DH,</I>
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926 <p n=>45</p>
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927 <I>minore che DG: $o$tentar&agrave; la rota inferiore $econdo la ragione di EH,
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928 ad ED; e la $uperiore $ecodo la ragione di DH ad ED: $o$tenta d&utilde; que
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929 la rota inferiore, maggior portione di pe$o: e la $uperiore minor porti
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930 one, che $e nel $ito orizontale fu$$ero. Il che $i hauca da mo$trare.</I>
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931 <FIG>
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932 <HEAD><I>PROPOSITIONE.</I>
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933 V.</HEAD>
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934 <p>Data la rota che aff&otilde;di in c&otilde;cauita $otto il piano oriz&otilde;
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935 <p n=>46</p>
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936 tale: e data qual si uoglia grauezza: ritrouare in vn rag-|
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937 gio la di$tanza oltre di cui detta grauezza appe$a, $ol-
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938 leui detta rota.
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939 <FIG>
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940 <HEAD><I>Dimostratione.</I></HEAD>
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941 <p><I>Sia la linea del piano orizontale ABCD: la concauit&agrave; in e$$a BE
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942 C: la rota che affondi BCF: la grauezza data G. $i cerca in vn raggio
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943 della rota, ponto oltre di cui $o$pe$a la G, $olleui detta rota. Sia il cen-
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944 tro H: la linea del raggio prodotto HFI: qual $ia parallela all'ori-
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945 zonte: e dal p&otilde;to C, $i tiri la CK perpendicolare che affr&otilde;ti la HF, in K:
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946 e la ragion c'ha la grauezza G al pe$o della rota, habbia HK a KI: &egrave;
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947 manife$to perche KC, &egrave; perdendicolare del $o$tenimento, che dal ponto I
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948 la grauezza G, fa equipondio allarota. e che da ogni ponto oltre, la $ol
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949 leui, il che $i cercaun.</I>
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950 <p n=>47</p>
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951 <HEAD>TAGLIA.
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952 <I>SVPPOSITIONE.</I></HEAD>
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953 <p>Svpponiamo la taglia c'habbia in $e una, o pi&ugrave;
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954 girelle, o sia in vno o pi&ugrave; ordini. Et delle taglie, $ta
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955 bile diciamo, il cui collo sia legato ad vn termine: mo-
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956 bile il cui collo sia legato al pe$o. Et altrim&etilde;te mobile la
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957 guidata da vna potenza, e che ad vn capo di e$$a $ia attac
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958 cato il pe$o. In oltre $upponiamo della corda auuolta il
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959 capo andare, o alla taglia, o ad'vn termine fi$$o, o a pof-
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960 $anza, &ograve; a pe$o.
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961 <HEAD><I>POSITIONE</I>
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962 I.</HEAD>
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963 <p>Poniamo della girella a cui sia auuolta corda data
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964 a pesi, &amp; a po$$anze, mentre detta girella non volta il mo
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965 mento de capi e$$ere vguale.
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966 <HEAD>II.</HEAD>
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967 <p>Ma $e la girella volta, il momento di quella corda e$-
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968 $er maggiore, ver$o di cui volta.
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969 <HEAD>III.</HEAD>
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970 <p>E poniamo nelle girelle, di po$$anze e pe$i vguali,
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971 li momenti e$$ere vguali.
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972 <p n=>48</p>
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973 <HEAD><I>PROPOSITIONE.</I>
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974 I.</HEAD>
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975 <p>Se delli due capi della girella, l'vna $o$tenti pe$o, l'al
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976 tro $ia dato a po$$anza: la po$$anza del capo $ar&agrave; di mo
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977 mento eguale al pe$o. e la po$$anza della taglia $o$tenta
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978 r&agrave; il doppio.
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979 <HEAD><I>Dimo$tratione.</I></HEAD>
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980 <p><I>Sia la taglia AB: li capi della fune auuolta</I>
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981 <FIG>
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982 <I>A C, B D: dequali A C, $o$tenti il pe$o C: e B
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983 D, $ia dato alla po$$anza in D: dico che la
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984 po$$anza in D &egrave; di momento eguale al pefo: e
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985 che la po$$anza in E, $o$tenta il doppio. Si
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986 mo$tra: e prima che'l momento di D, $ia v-
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987 guale al momento di C. &egrave; manife$to: perche
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988 $e l'vn di loro fu$$e maggiore, la girella volte
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989 rebbe ver$o detto momento: Jl che &egrave; contro
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990 il $uppo$to. Dico hora che la<*>$$<*>za della
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991 taglia $ia doppia del pe$o: percioche e$$endo
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992 la po$$anza di D, equiualente al pe$o C: ambi
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993 C e D, $ono il aoppio di e$$o C: ma la po$s&atilde;za in
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994 E, in quanto $o$tiene, &egrave; vguale ad ambi: dun-
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995 que &egrave; doppia di vn di loro. Ha$$i dunque il
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996 propo$to, che la po<32>anza D, $ia vguale al mo
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997 mento di C: e che la E, $o$tenti il doppio di e$$o.</I>
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998 <p n=>49</p>
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999 <HEAD><I>PROPOSITIONE.</I>
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1000 II.</HEAD>
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1001 <p>Se li due capi di girella mobile, $iano raccomanda-
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1002 ti a due po$$anze: $o$tentar&agrave; cos&igrave; l'vna, come l'altra po$
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1003 $anza, la met&agrave; del pe$o.
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1004 <HEAD><I>Dimo$tratione.</I></HEAD>
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1005 <p><I>Sia la taglia A B, a cui $ia attaccato il pe</I>
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1006 <FIG>
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1007 <I>$o <*>: li due capi della corda auuolta alla gi-
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1008 rella A D, B E: le po$$anze in D, et E: dico
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1009 che cos&igrave; l'vna, come l'altra po$$anza $o$ten
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1010 ta la met&agrave; del pe$o. $i mo$tra: percioche $tan&shy;
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1011 do la girella s&etilde;za voltare, $ec&otilde;do il $up: $ara
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1012 di con$eguenza il momento dell'vn capo v-
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1013 guale al momento dell'altro: e perci&ograve; le po$
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1014 $anze anco eguali. e perche ambe $o$tenta-
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1015 no il pe$o C: e le po$$anze, in quanto $o$ten-
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1016 gono, $ono eguali alli pe$i. $ono dunque am
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1017 be eguali al pe$o C: e perci&ograve; diui$amente l'v
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1018 na e l'altra $ar&agrave; la met&agrave; di detto pe$o-
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1019 al che $i hauea da mo$trare.</I>
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1020 <foot><I>G</I></foot>
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1021 <p n=>50</p>
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1022 <HEAD><I>Appendice,</I></HEAD>
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1023 <p>E perci&ograve; anco se l'vn capo sia raccomandato ad vn
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1024 termine fi$$o, l'altro a po$$anza: $o$terr&agrave; la po$$anza la
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1025 met&agrave; del pe$o.
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1026 <p><I>Percioche mutato il termine in un'altra po$$anza: la po$$anza $uppo
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1027 $ta $o$terr&agrave; l'i$te$$a altra quantit&agrave; di pe$o che prima.</I>
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1028 <HEAD><I>PROPOSITIONE.</I>
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1029 III.</HEAD>
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1030 <p>Delle corde, che dalla taglia $u
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1031 periore, &amp; dalla po$$anza alla ta
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1032 <FIG>
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1033 glia inferiore peruengono: cia$cu
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1034 na $o$tiene egual parte dipe$o.
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1035 <HEAD><I>Dimo$tratione.</I></HEAD>
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1036 <p><I>Sia la taglia $uperiore A B: l'inferiore
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1037 C D: la corda auuolta no tata con l'i$te<32>e let
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1038 tere: e di lei l'vn termine vada a $o$tenere
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1039 la taglia inferiore in E: l'altro $ia dato al-
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1040 la po$$anza in F. Dico che cia$cuna corda
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1041 $o$tiene egual parte di pe$o. Si mo$tra:
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1042 perche $tando la girella A B, il momento
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1043 del capo B D &egrave; eguale al momento del ca-
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1044 po A E: e del capo C F, al capo B D, per
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1045 la girella C D: $ono d&utilde;que tutte di momen&shy;
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1046 to eguali: perci&ograve; cia$cuna $o$tentar&agrave; e-
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1047 gual parte di pe$o. e $e il capo A E non fu$
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1048 $e ligato alla taglia, ma ad altro termine, $a
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1049 rebbe l'i$te$$o, ma il numero delle corde di
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1050 vna meno. Il che $i hauea da mo$trare.</I>
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1051 <p n=>51</p>
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1052 <HEAD><I>PROPOSITIONE.</I>
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1053 IIII.</HEAD>
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1054 <p>Se l'vn capo della fune auuolta
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1055 <FIG>
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1056 a girelle, $ia raccomandato alla ta
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1057 glia $uperiore: il pe$o $o$tenuto
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1058 &egrave; di$tribuito in parti di numero
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parents:
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1059 pare.
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1060 <HEAD><I>Dimostratione.</I></HEAD>
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1061 <p><I>Sia la taglia inferiore e mobile AC B D:
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1062 la $uperiore E F G H: la fune auuolta nota,
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1063 ta, c&otilde; l'iste$$e lettere: il termine del capo C 1,
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parents:
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1064 attaccato alla taglia $uperiore, s'int&etilde;da e$$e
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1065 re in I: l'altro termine raccomandato alla
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1066 po$$anza s'intenda e$$ere o in K del capo B
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1067 K, che vien dalla taglia inferiore, o in L, del
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1068 capo G L, che vien dalla taglia $uperiore.
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1069 Dico che, e nell'vno, e nell'altro modo, il pe
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1070 $o &egrave; di$tribuito in parti di numero pare.
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1071 Si mo$tra: percioche venendo alla girella
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1072 C D due corde, l'vna da taglia, l'altra da-
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1073 girella E F: $aranno detti capi di momen&shy;
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1074 ti eguali: perche $i pone la girella non vol
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parents:
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1075 tare. $imilmente perche alla girella A B
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1076 vengono due corde, l'vna dalla girella EF,
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1077 che &egrave; la corda E A, l'altra dalla po<32>anza
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1078 K, che &egrave; la corda KB: $aranno dette corde
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1079 di mom&etilde;ti eguali. ma la DF, &egrave; di momento
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parents:
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1080 eguale alla A E, e alla B K: $ono dunque
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1081 tutte tra di loro di momento eguale: e
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1082 $ono di numero pare: percioche a cia-</I>
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1083 <foot><I>G</I> 2</foot>
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1084 <p n=>52</p>
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1085 <I>$cuna girella ne vengono due. perche dunque il pe$o &egrave; $o$tenuto da
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parents:
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1086 dette corde di mom&etilde;to eguale: perci&ograve;, mentre l'vn capo $ia attac cato
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parents:
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1087 alla taglia $uperiore, l'altro dato alla po$$anza, il mom&etilde;to del pe$o &egrave; di
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parents:
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1088 $tribuito in parti di numero pare: ne altro auuiene, $e la po<32>anza $ia in
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1089 L, nel capo, che viene dalla taglia $uperiore: percioche il numero del
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1090 le corde, che alla taglia inferiore peruengono &egrave; l'i$te$$o.</I>
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1091 <HEAD><I>Appendice. I</I></HEAD>
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1092 <p>Et &egrave; manife$to, che po$ta vna girella meno nella ta-
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1093 glia $uperiore, $i $o$terr&agrave; dalla po$$anza l'i$te$$o che $e
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1094 fu$$ero le girelle $uperiori di numero eguale alle in-
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parents:
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1095 feriori, &egrave; che per detta girella aggiunta, si muta $o
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parents:
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1096 lamente l'un momento nell'altro di $pezie contraria.
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1097 <HEAD><I>PROPOSITIONE.</I>
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1098 V.</HEAD>
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1099 <p>Se l'vn capo della fune auuolta a girelle, $ia racco-
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1100 mandato alla taglia inferiore: il pe$o $o$tenuto &egrave; di$tri
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1101 buito in parti di numero $pare.
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1102 <HEAD><I>Dimo$tratione.</I></HEAD>
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1103 <p><I>Sia la taglia $uperiore e stabile AB, CD: l'inferiore e mobile E-
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1104 F, G H: la fune auuolta notata con l'i$te$$e lettere: e di e$$a l'vn ter-
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1105 mine I, che &egrave; del capo C I, $ia attaccato alla taglia inferiore: etil termi
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1106 ne K, del capo E K, rac comandato alla po$$anza in K: dico che'l pe$o &egrave;</I>
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1107 <p n=>53</p>
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1108 <I>di$tribuito in parti dinumero $pare. Si</I>
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1109 <FIG>
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1110 <I>mo$tra: percioche vengono due capi dalla
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1111 girella C D, alla taglia inferiore, e due
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1112 dalla A B, e $imilmente da qual $i voglia
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1113 altra girella: $ono dunque li capi, che dal
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1114 le girelle alla taglia vengono, di numero
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1115 pare. et euui in oltre il capo della po$$an
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1116 za: $ono dunque tutti di numero $pare.
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1117 e $ono, per quel che $i &egrave; detto nelle prece
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1118 denti, tutte di momento eguale: dunque
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1119 il pe$o &egrave; di$tribuito in parti di numero
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1120 $pare. Jl che $i hauea da mo$trare.</I>
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1121 <HEAD><I>Appendice. I.</I></HEAD>
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1122 <p>Et &egrave; manife$to, che aggionta
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1123 alla taglia $uperiore vna girel-
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1124 la, si commuta $olamente il mo-
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1125 mento della po$$anza, in mo-
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1126 mento di $pezie contraria.
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1127 <HEAD><I>Appendice. II.</I></HEAD>
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1128 <p>E raccogliamo, che ligato l'vn
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1129 capo alla taglia $uperiore, puote
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1130 $tar detta taglia con vna girella
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1131 meno: e ligata all'inferiore con
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1132 vna girella pi&ugrave;.
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1133 <p n=>54</p>
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1134 <HEAD><I>PROPOSITIONE.</I>
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1135 VI</HEAD>
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1136 <p>Se vn capo della taglia $upe
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1137 <FIG>
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1138 riore sia raccomandato ad vn
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1139 termine fi$$o: $ar&agrave; il pe$o di$tri
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1140 buito in parti di numero pare.
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1141 <HEAD><I>Dimo$tratione.</I></HEAD>
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1142 <p><I>Sia la taglia $uperiore A B C D, l'-
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1143 inferiore E F G H: la fune auuolta a gi
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1144 rellenotata con l'i$te$$e lettere: di cui
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1145 il capo D I dalla girella C D della taglia
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1146 $uperiore $ia raccomandato ad I termi
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1147 ne fi$$o: &amp; il capo F K, dalla girella E F,
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1148 della taglia inferiore, raccomandato al
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1149 la po$$anza in K. Dico che'l pe$o &egrave; di$tri
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1150 buito in parti di numero pare. Si mo-
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1151 $tra: percio che venendo alla taglia infe
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1152 riore le corde $olo delle girelle, &amp; da cia
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1153 $cuna girella due corde, quali tutte $i &egrave;
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1154 mo$trato che $o$tentino egual momento:
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1155 $ar&agrave; il pe$o di$tribuito in corde di nume
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1156 ro pare, che egualmente $o$tentano: e
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1157 perci&ograve; $ar&agrave; di$tribuito in dette parti. Il
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1158 che $i hauea da mo$trare.</I>
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1159 <HEAD><I>Correlario. I.</I></HEAD>
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1160 <p>E manife$to dunque che li
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1161 gatoil capo di $opra alla taglia
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1162 <p n=>55</p>
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1163 inferiore, il pe$o &egrave; di$tribuito in parti di numero $pare,
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1164 et comunque altrimente, in parti di numero pare.
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1165 <HEAD><I>Correlario. II.</I></HEAD>
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1166 <p>Et attaccato il capo di girella inferiore alla taglia $u
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1167 periore, o &agrave; qual si voglia termine fi$$o: che la taglia in
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1168 feriore habbia vna girella pi&ugrave;.
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1169 <HEAD><I>PROPOSITIONE.</I>
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1170 VII.</HEAD>
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1171 <FIG>
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1172 <p>Se'l pe$o sia mo$$o con ta
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1173 glie, quanto il pe$o &egrave; moltepli
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1174 ce della po$$anza $o$tenente,
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1175 tanto lo $patio, che detta po$-
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1176 s&atilde;za camina, &egrave; molteplice del
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1177 lo $patio caminato dal pe$o.
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1178 <HEAD><I>Dimostratione.</I></HEAD>
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1179 <p><I>Sia la girella dellataglia $uperiore
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1180 A B: della inferiore nella prima po-
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1181 $itione $ia C D: e lapo$$anza che $o$tie
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1182 ne il capo $ia in E: della $econda po$i
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1183 tione $ia in G H, e la po$$anza in I. Di
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1184 co che lo $patio caminato dalla taglia
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1185 mobile e pe$o, &egrave; talparte dello $patio
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1186 E I, qual la po$$anza $o$tenente in E
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1187 &egrave; parte del pe$o. Si mo$tra: perche
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1188 quante $ono la corde, che alla taglia</I>
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1189 <p n=>56</p>
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1190 <I>inferiore peruengono, $econdo tal numero la po$$anza che $o$tiene &egrave;
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1191 parte del pe$o: e perche nel mouimento della taglia cia$cuna corda
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1192 $i abbreuia egualmente, portata C D, in H G: le C G, D H parti del
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1193 la corda auuolta, quante $i $iano, pigliate in$ieme, $arano di lunghezza
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1194 tanto molteplici dello $patio caminato, quanto &egrave; il numero delle cor
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1195 de. ma la corda E A B D C F, &egrave; vguale alla I A B H G: dunque tol
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1196 tone di commune la F G H B A E, re$ta le E I, eguale alla G C D H:
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1197 e percto E I, $ar&agrave; altre tanto molteplice dello $patio caminato, quan
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1198 to erano le corde C G, D H. ci&ograve; &egrave; il pe$o tutto del pe$o da vna corda
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1199 $ostenuto. Il che $i hauea da mo$trare.</I>
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1200 <HEAD><I>PROPOSITIONE.</I>
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1201 VIII.</HEAD>
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1202 <p><I>Problema. I.</I>
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1203 <p>Data qual si voglia grauezza, e po$$anza: ritroua
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1204 re il minor numero di girelle nella taglia, con quali
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1205 la data po$$anza moua il dato pe$o.
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1206 <HEAD><I>Dimo$tratione.</I></HEAD>
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1207 <p><I>Sia la data grauezza A, la po$$anza B: a cui $i pigli vn pe$o e-
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1208 quiualente C: e moltiplichi$i C, $in che la prima volta ecceda la
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1209 grauezza A, il che $ia per il numero D. $e dunque D &egrave; pare piglin
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1210 $i nella taglia inferiore altre tante girelle, quante vnit&agrave; $ono nella
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1211 inet&agrave; del numero: &egrave; manife$to che la po$$anza mouer&agrave; il pe$o con le
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1212 date girelle: ma $e D $ia $pare, toltane vnit&agrave;, piglin$i girelle quan
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1213 te vnit&agrave; $ono nella met&agrave; del re$to, e lighe$i vn delli capi alla taglia:
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1214 &egrave; manifesto $i milmente che mouer&agrave; la po$$anza la data grauezza.
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1215 Il che $i cercaua.</I>
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1216 <p n=>57</p>
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1217 <HEAD><I>PROPOSITIONE.</I>
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1218 VIII.</HEAD>
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1219 <p><I>Problema. II.</I>
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1220 <p>Data qual $i voglia velocit&agrave;, e data la tardit&agrave; della
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1221 po$sanza: applicar o vna taglia di pi&ugrave; girelle, o pi&ugrave; ta
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1222 glie di vna girella, $i che la po$$anza moua il dato pe-
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1223 $o in velocit&agrave; magior della data.
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1224 <HEAD><I>Dimo$tratione.</I></HEAD>
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1225 <p><I>Pigli$i lo $patio che nel dato tem-</I>
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1226 <FIG>
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1227 <I>po camini la po$$anza: e lo $patio che
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1228 vogliamo che la co$a camini, e $i mol
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1229 tiplichi il minore fin che la prima vol
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1230 ta auanzi, e quanto que$to &egrave; molte-
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1231 plice, tante corde $iano nella taglia $u
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1232 periore, pigliando la met&agrave; di girelle $e
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1233 $ia pare, &amp; $e $ia $pare, ligando vn ca
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1234 po ad e$$a taglia $uperiore. Ligato dun
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1235 que il pe$o ad vn capo, la po$$anza, che
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1236 tira la taglia, tirer&agrave; anco il pe$o: e ca-
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1237 miner&agrave; lo $patio moltiplice al moui-
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1238 mento di e$$a po$$anza. Ma, $e vo
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1239 gliamo far ci&ograve; con pi&uacute; taglie di vna gi
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1240 rella, radoppi$i lo $patio, e di nuouo il
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1241 fatto dal radoppiamento $i radopp&yuml;: e
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1242 ci&ograve; $i torni a fare, si che l'ultimo radop
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1243 piamento auanzi lo $patio maggiore.
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1244 Se dunque, quante volte $i &egrave; radoppia-
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1245 to, tanto numero ditaglie $i pigli, si mo
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1246 uer &agrave; ispe$o $econdo la ragion del radop
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1247 piamento dello $patio, e perci&ograve; $i mo-
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1248 uer&agrave; con maggior velocit&agrave; della data.</I>
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1249 <foot><I>H</I></foot>
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1250 <p n=>58</p>
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1251 <HEAD>ROTE MO
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1252 TIVE.</HEAD>
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1253 <FIG>
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1254 <HEAD><I>SVTPOSITION.</I>
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1255 I.</HEAD>
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1256 <p>Svpponiamo il mouimento di rote in a$$i
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1257 che $tanno co'l toccamento, communicar$i l'vna
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1258 all'altrail mouimento: e che'l momento della po$$an-
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1259 za $ia per linea chefaccia angolo retto co'l raggio di
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1260 e$$arota: e de momenti altri e$$er concorrenti, altri
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1261 contrarij.
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1262 <HEAD><I>DEFINITION.</I>
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1263 I.</HEAD>
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1264 <p>Concorrenti momenti diciamo quelli, che portan&shy;
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1265 do ver$o l'i$te$$a parte, $i accre$cono.
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1266 <HEAD>II.</HEAD>
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1267 <p>Contrarij quelli, che s'impedi$cono portando in
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1268 contrario.
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1269 <HEAD><I>POSITION.</I>
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1270 I.</HEAD>
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1271 <p>Poniamo, po$$anze eguali in circonferenze direte
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1272 eguali, hauer momenti eguali.
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1273 <p n=>59</p>
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1274 <HEAD><I>POSITION.</I>
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1275 II.</HEAD>
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1276 <p>Et in rote ineguali hauer momento ineguale, $econ
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1277 do la ragion de $emidiametrj.
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1278 <HEAD>III.</HEAD>
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1279 <p>E gli momenti contrarij, per quanto $i annullano, l'
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1280 vno e$$ere eguale all'altro.
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1281 <HEAD><I>PROPOSITION.</I>
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1282 I.</HEAD>
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1283 <p>Se quante $i voglia rote, vna per a$$e, $i tocchino:
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1284 e po$te le po$$anze l'vna nella circonferenza della pri
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1285 ma, e l'altra dell'vltima, $i rattengano: $aranno le po$
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1286 $anze egualj.
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1287 <FIG>
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1288 <foot>H 2</foot>
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1289 <p n=>60</p>
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1290 <HEAD><I>Dimostratione.</I></HEAD>
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1291 <p><I>Siano quante $i voglia rote ne gli a$$i A, B, C, che $i tocchino: ci&ograve;
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1292 &egrave; che la A tocchi la B nelponto D: e la B tocchila C nel ponto E:
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1293 &amp; intenda$i nella circonferenza di A e$$er la potenza F: e nella cir
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1294 conferenza di C la potenza G: che l'una rattenga l'altra. Dico che
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1295 le potenze $ono eguali. Si mo$tra: percio che la po<32>anza in F, &egrave; dell'-
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1296 i$te$$o momento, che $e fu$$e in D, dell'i$te$$a rota A: ma il ponto
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1297 D, &egrave; ponto commune a due rote: e la po$$anza in D della rota B,
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1298 &egrave; quanto fu$$e in E: $ar&agrave; dunque la po$$anza in F l'i$te$$o che $i fu$$e
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1299 in E: perche d&utilde;que la po$$anza in F $i annulla con la po$$anza in G, $o
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1300 no li loro momenti eguali. Ma le po$$anze che $ono in un'i$te$$a rota
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1301 di momenti eguali, $ono eguali: dunque la po$$anza in F &egrave; uguale alla
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1302 po$$anza in G. Jl che $i hauea da mo$trare.</I>
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1303 <HEAD><I>PROPOSITION.</I>
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1304 II.</HEAD>
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1305 <p>Delle due rote in vno a$$e la po$$anza, che fa egual
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1306 momento nella rota magiore &egrave; di valor minore: e nel
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1307 la minore &egrave; di valor maggiore, nella ragione de $emi
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1308 diametri reciproca.
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1309 <HEAD><I>Dimostratione.</I></HEAD>
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1310 <p><I>Siano $u l' a$$e A le rote A B, A C: &amp; intenda$i la po$$anza B,
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1311 in circonferenza della rota maggiore, hauere egual momento alla
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1312 po$$anza C in circonferenza della rota minore. Dico che la po$$an-
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1313 za B &egrave; minore della po$$anza C, $econdo la ragione di C A ad A B.
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1314 Si mo$tra: intenda$i nell circonferenza di A C e$$er po$$anza eguale
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1315 a B, che $ia D: $ar&agrave; il momento di B al momento di D, nella ragion</I>
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1316 <p n=>61</p>
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1317 <FIG>
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1318 <I>della linea dritta B A alla D A: ma il momento di B, &egrave; uguale al
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1319 momento di C: dunque il momento di C al momento di D, &egrave; come
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1320 B A ad A D. Se dimque le po$$anze dell'i$te$$a rota $ono tra di loro
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1321 nella ragione delli momenti: $ar&agrave; di con$eguenza la po$$anza in D
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1322 alla po$$anza in C, come il $emidiametro D A, al $emidiame-
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1323 tro A B, e del diametro tutto a tutto. Il che $i hauea da mo-
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1324 $trare.</I>
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1325 <p n=>62</p>
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1326 <HEAD><I>PROPOSITION.</I>
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1327 III.</HEAD>
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1328 <p>Se le rote, po$te a due in cia$cun a$$e, $i tocchino:
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1329 e le po$$anze, po$te l'vna nella prima, l'altra nell'vl-
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1330 tima rota, $i rattengano: $ar&agrave; la ragion dell'vna po$$an
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1331 za all'altra l'i$te$$a, che la ragion compo$ta delli $emi
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1332 diametri, che $ono $u l'i$te$$o a$$e, pigliate reciproca-
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1333 mente.
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1334 <HEAD><I>Dimo$tratione.</I></HEAD>
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1335 <p><I>Siano $u l'a$$e A, le due rote A B, A C: e $u l'a$$e D, le rote D C
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1336 D F: &amp; intenda$i la rota D C, e$$er toccata dalla A C nel ponto C:
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1337 c l'una po$$anza e$$ere in B l'altra in E. Dicoche la po$$anza in B,</I>
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1338 <FIG>
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1339 <I>alla po$$anza in F ha la ragion compo$ta delle ragioni di F D a D C,</I>
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1340 <p n=>63</p>
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1341 <I>e di C A ad A B, che $ono le ragioni de $emidiametri reciprocamen
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1342 te pigliati. Si mo$tra: percioche e$$endo il momento in B uguale al
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1343 momento in C, perche $ono in vno i$te$$o a$$e: &amp; il momento in C al
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1344 momento in F, per l'iste$$a ragione: &amp; &egrave; la po$$anza in F, alla
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1345 po$$anza in C, come il diametro C D a D F: e la po$$anza in C,
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1346 alla po$$anza in B, co me B A ad A C. Dunque la po$$anza in F alla
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1347 po$$anza in B, ha la ragion compo$ta di C D a D F c di B A ad A C,
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1348 che &egrave; la ragion compo$ta delle ragioni de diametri reciprocamente
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1349 pigliati. Jl che $i hauea da mo$trare.</I>
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1350 <HEAD><I>DEFINITIONE.</I></HEAD>
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1351 <p>Momento della rota diciamo, il momento del pon
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1352 to po$to nella circonferenza di e$$a rota.
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1353 <HEAD><I>PROPOSITION.</I>
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1354 IIII.</HEAD>
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1355 <p>Se in vna congiogation di rote ineguali, o in pi&ugrave;,
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1356 che la minor dell'vna congiogatione tocchi la mag-
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1357 gior dell'altra, $i ponga la po$$anza in vna di dette
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1358 rote: $ar&agrave; il momento dell'vltima minor rota, maggior
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1359 del momento della prima maggior rota, $econdo la ra&shy;
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1360 gion compo$ta delli diametri. e la velocit&agrave; $ar&agrave; mino
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1361 re, $econdo l'i$te$$a ragion de diametri.
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1362 <HEAD><I>Dimo$tratione.</I></HEAD>
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1363 <p><I>Siano le congiogationi di rote, de quali gli a$si $iano A e B: &amp; in|-
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1364 tenda$i $u l'a$$e A e$$er la rota maggiore A C, e la minore A D,</I>
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1365 <p n=>64</p>
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1366 <FIG>
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1367 <I>e $u l' a$$e B, e$$er la maggiore D B, e la minore B E: e $ia il contat
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1368 to della minore di vn'ordine, con la maggiore dell'altro, il ponto D:
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1369 e $upponga$i prima la po$$anza por$i nella cir conferenza di A C.
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1370 Dico che'l momento della rota A D, &egrave; maggiore del momento di
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1371 A C, secondo la ragione della linea C A ad A D. Si mo$tra: per
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1372 cioche po$ta in D una po$$anza di mom&etilde;to eguale alla po$$anza in C,
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1373 $ar&agrave; detta po$$anza in D, maggiore, che la po$$anza in C: ma ilmo
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1374 mento della rota, oue &egrave; po$ta la po$$anza, &egrave; uguale ad e$$a po$$an-
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1375 za: $ar&agrave; dunque il mom&etilde;to della rota A D maggiore che della rota
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1376 A C $econdo la ragion de diametri: que$to in una congiogatione
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1377 &amp; in pi&ugrave;: per che il momento della circonferenza di A D &egrave; l'i$te$$o
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1378 che della circonferenza di B D, per lo contatto, che fa communi-
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1379 canza: ma il momento della circonferenza di B E, &egrave; di forza
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1380 maggiore che di B D $ec&otilde;do la ragione del diametro, B D a B E:
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1381 dunque fatta compo$itione de ragioni il momento della circonferen
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1382 za di B E, &egrave; maggiore del momento della circonferenza di C A $e
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1383 condo la ragion compo$ta di B D a B E, e di C A ad A D. Il che
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1384 $i hauea da mo$trare.</I>
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1385 <p><I>Dico che la uelocit&agrave; &egrave; minore nella i$te$$a ragione: il che &egrave; mani-</I>
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1386 <p n=>65</p>
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1387 <I>fe$to: perc&itilde;oche la velocit&agrave; delle rote, che nell'i$te$$o tempo fini$cono
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1388 il circuito, &egrave; proportionale alle circonferenze di e$$e rote: e le circon
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1389 ferenze $ono di quantit&agrave; proportionale alli diametri. Sono dunque le
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1390 velocit&agrave; delle rote proportionali alli diametri. Jl che $i hauea da
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1391 mo$trare.</I>
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1392 <HEAD><I>PROPOSITION.</I>
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1393 V.</HEAD>
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1394 <p>Date due po$$anze di momento contrario, l'vna mi
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1395 nore, el'altra maggiore: e data la ragione dell'vna al-
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1396 l'altra delle due rote congiogate: ritrouar il minor nu-
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1397 mero de congiogationi, $iche la data po$$anza minore
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1398 vinca la maggiore.
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1399 <FIG>
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1400 <foot>I</foot>
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1401 <p n=>96</p>
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1402 <HEAD><I>Dimostratione.</I></HEAD>
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1403 <p><I>Siano le date po$$anze di momento contrario A, B: De quali A
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1404 $ia la maggiore, c B la minore.: la ragion delle rote congiogate $ia di
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1405 C a D: $i cerca il minor numero de congiogationi, $iche la po$$anza
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1406 B minore vinca la A maggiore. Piglin$i nella ragione di C a D con
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1407 tinuamente le C, D, E, F: $iche la C ad F habbia maggior ragione
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1408 che l' A a B: &amp; eguale di numero all'interualli de termini $i piglino
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1409 le congiogationi di rote G, H, I: e $iano $u l'a$$e G, le rote G K, G
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1410 L, $u l'a$$e H le rote H L, H M: e $u l'a$$e I le rote M I, I N. E ma
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1411 nife$to che'l momento della po$s&atilde;za in K, al momento $uo in N, ha la
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1412 ragion compo$ta delle ragioni de $emidiametri: e perci&ograve; po$ta la po$-
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1413 $anza maggiore A in N: e la minore B in K: $ara il momento della B
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1414 in K, maggiore che'l momento dell' A in N. Il che $i hauea da
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1415 trouare.</I>
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1416 <HEAD><I>PROPOSITION.</I>
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1417 VI.</HEAD>
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1418 <p>Data qual$ivoglia tardit&agrave; di po$$anza, &amp; qual$ivo
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1419 glia velocit&agrave;: e data la ragion de diametri delle rote c&otilde;
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1420 giogate: ritrouar vn minimo numero de congiogatio
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1421 ni, $i che la data po$$anza moua la co$a con velocit&agrave;
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1422 maggiore della data.
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1423 <HEAD><I>Dimostratione.</I></HEAD>
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1424 <p><I>Sia la po$$anza tarda A, la veloce B, lo $patio caminato da A in
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1425 vn dato tempo $ia C, lo C caminato da B nell'i$te$$o tempo $ia D: la
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1426 ragion de diametri congiogati $ia di E, ad F: bi$ogna ritrouare il</I>
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1427 <p n=>67</p>
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1428 <FIG>
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1429 <I>minimo numero de c&otilde;giogationi, col quale la tarda A moua con ve
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1430 locit&agrave; maggior che'l B. Piglin$i le E, F, G, continuate nella ragion
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1431 de diametri, che la prima volta l'interuallo della prima all'vltima
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1432 dico di G ad E, $ia maggiore che di C a D: e quanti interualli $ono il
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1433 E, F, G: tante congiogationi di rote $i piglino nella i$te$$a ragione: l'a
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1434 $e de quali $iano H, I: e nello a$$e H, la minor rota $ia H K, la maggio
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1435 re H L: e nell'a$$e I laminore I L, la maggiore L M. il contatto del
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1436 l'vna congiogatione all'altra il ponto L: &egrave; manife$to che la veloci-
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1437 t&agrave; del ponto M, alla velocit&agrave; del ponto K, &egrave; compo$ta della ragion del
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1438 li diametri M I, ad I L, &amp; H L ad H K: che &egrave; l'i$te$$a, che di G ad E:
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1439 ma G ad E, &egrave; di maggior interuallo che di D a C. D&utilde;que, po$ta la po$$an
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1440 za tarda in K, la co$a mo$$a con la rirconferenza M, $i mouer&agrave; c&otilde; mag
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1441 gior velocit&agrave; della data. Il che $i hauea da trouare.</I>
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1442 <foot><I>I</I> 2</foot>
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1443 <p n=>68</p>
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1444 <HEAD><I>MOMENTI ACQVISTATI.</I></HEAD>
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1445 <p>Poniamo degli momenti, altri e$$er intrin$echi: al
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1446 tri acqui$tati, &amp; altri mi$ti: &amp; intrin$echi quelli, che
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1447 non da mouimento precedente dipendono: come $ono
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1448 gli mouimenti delle grauezze in gi&ugrave;, e del corpo leggiero
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1449 dentro l'humor pi&ugrave; graue in s&ugrave;. Acqui$tati quelli, che $e-
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1450 guono l'impre$sion fatta da precedente mouimento: come
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1451 il mouimento della co$a lanciata, che $egue il mouim&etilde;to
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1452 del braccio, o della corda. Mi$ti, come il mouimento delle
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1453 grauezze dopo l'hauer dato principio a mouer$i: per il che
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1454 veggiamo li pe$i di vicino la$ciati, mouer$i con minor mo-
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1455 mento, che la$ciati di lontano: e molte co$e portate dalla
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1456 propria grauezza nell'aria penetrar $otto l'accqua, con-
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1457 tro di quel che porta l'intrin$eco momento: onde dopo
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1458 l'e$$ere affondate da $e $te$si ritornar &aacute; galla. Et il momen&shy;
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1459 to intrin$eco e$$er l'i$te$$o $empre. l'acqui$tato, mancando
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1460 la cau$a di poner$i, e con il tempo, e dall'impedimento che
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1461 le faccia re$i$tenza. <I>CVGNO.</I>
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1462 <p>Il cugno perco$$o, con$iderato in vn modo, rappre$enta
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1463 un piano inchinato, che $i $pinga $otto il pe$o. Et altrimen
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1464 te rappre$enta due leue, che nelle loro $tremit&agrave;, facciano
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1465 l'vna all'altra $ottoleua, &amp; habbiano il pe$o tra la po$$anza,
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1466 e'l $ottoleua. Et altrimente rappre$enta leua nel cui $tremo
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1467 $ia il pe$o, &amp; il $ottoleua tramezzo. <I>VITE E CHIOCCIA.</I>
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1468 <p>La vite, o chioccia rappre$enta vno o pi&ugrave; piani auuolti
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1469 ad vn fu$ello. Sono e ma$chia, e femina: de quali vna $tan&shy;
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1470 do ferma, l'altra che gira $o$tiene il pe$o. acqui$ta dunque for
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1471 za, $ec&otilde;do la detta inchinazione, e $econdo la lunghezza del
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1472 raggio che $e le accompagna. Vite perpetua diciamo vn
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1473 <*>ympano con denti a vite, che girando tocchi rota dentata.
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1474 Per il che accre$ce la forza, e per la proprieta della vite,
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1475 e della congiogatione delle rote.
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1476 <HEAD><I><*>L FINS.</I></HEAD>