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annotate texts/XML/archimedes/it/galla_perig_it_1590.xml @ 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
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1 <?xml version="1.0"?>
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2 <archimedes>
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3 <info>
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4 <author>Gallaccini, Teofilo</author>
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5 <title>Perigonia, o vero degli angoli (Ms. L. IV. 5 della Biblioteca degli Intronati di
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6 Siena, cc. 1r. - 86r.)</title>
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7 <date>ca. 1590-1598</date>
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8 <place>Siena</place>
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9 <translator>A cura die Annalisa Simi</translator>
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10 <lang>it</lang>
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11 <cvs_file>galla_perig_it_1590.xml</cvs_file>
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12 <echodir>/permanent/echo/biblioteca_siena/galla_perig_it_1590</echodir>
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13 </info>
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14
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15 <text>
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16 <body>
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17 <chap>
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18 <pb pagenum="folio 1r"></pb>
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19 <p type="head">
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20 <s>PERIGONIA, O VERO <lb></lb>DEGLI ANGOLI. </s>
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21 </p>
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22 <p type="head">
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23 <s>Di Teofilo Gallaccini </s>
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24 </p>
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25 <p type="main">
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26 <s>Quantunque secondo l’opinione d’alcuni le Scienze Matematiche sieno di
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27 maniera dichiarate, che si mostrano non haver bisogno di più chiarezza; con
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28 tutto ciò, avendo soggette alla speculatione dello ‘ntelletto nostro son
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29 capaci d’altre considerationi; onde non sarà maraviglia se le medesime cose
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30 della Geometria secondo diversa consideratione benchè determinate pel
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31 diverso modo di considerare si facciano diverse. </s>
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32 <s>Onde l’Angolo essendo una particella di tutto l’obietto della Geometria,
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33 considerato da noi in varie maniere e secondo la diversa applicatione e
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34 secondo gli effetti diversi, diverrà diverso, dico (favellando
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35 filosoficamente) non nell’esser reale, ma nel formale; ed essendo diverso ci
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36 darà occasione di considerare di lui diverse positioni le quali si notaranno
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37 qui appresso.</s>
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38 </p>
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39 <p type="main">
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40 <s>Se l’angolo sia cosa reale o intelligibile o immaginaria. </s>
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41 <s>Questa si considerarà. Nel 1° cap. </s>
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42 </p>
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43 <p type="main">
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44 <s>L’angolo di quante maniere sia. Nel 2° cap. </s>
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45 </p>
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46 <p type="main">
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47 <s>Che cosa sia l’angolo ed in che cosa sia collocata l’essenza sua. Nel 3° cap.
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48 </s>
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49 </p>
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50 <p type="main">
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51 <s>Che per la varia division del cerchio si ritrovano tutte le specie degli
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52 angoli. Nel 4° cap. </s>
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53 </p>
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54 <p type="main">
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55 <s>D’altri tagliamenti del cerchio onde risultano diverse maniere d’angoli. Nel
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56 5° cap. </s>
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57 </p>
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58 <p type="main">
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59 <s>Se ogn’angolo sia divisibile. Nel 6° cap.</s>
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60 </p>
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61 <p type="main">
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62 <s>Se si dà l’angolo indeterminato sì come si dà il determinato. Nel 7° cap.
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63 </s>
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64 </p>
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65 <p type="main">
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66 <s>Se l’angolo si riduca al tutto alla pianezza e ugualità della linea retta o
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67 vero alla curvità della circolare. Nel 8° cap. </s>
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68 </p>
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69 <p type="main">
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70 <s>Per qual ragione alcune volte il cerchio sia detto tutto angolo. Nel 9° cap.
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71 </s>
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72 </p>
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73 <p type="main">
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74 <s>L’angolo a che serva nell’Universo. Nel 10° cap. </s>
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75 </p>
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76 <p type="main">
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77 <s>Di quale utilità e di che uso sia l’angolo nella Geometria. Nel cap. 11° </s>
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78 </p>
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79 <pb pagenum="folios 1v-2r"></pb>
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80 <p type="main">
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81 <s></s>
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82 </p>
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83 <p type="main">
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84 <s>Nell’Astronomia. Nel cap. 12° </s>
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85 </p>
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86 <p type="main">
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87 <s>Nella Prospettiva. Nel cap. 13° </s>
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88 </p>
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89 <p type="main">
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90 <s>Se ‘l colore è obietto proprio del vedere. Nel cap. 14° </s>
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91 </p>
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92 <p type="main">
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93 <s>L’uso degli angoli nella prospettiva appartenente agli specchi e a’
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94 reflettimenti de’ raggi del Sole. Nel cap. 15° </s>
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95 </p>
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96 <p type="main">
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97 <s>Nella Prospettiva scenografica. Nel cap. 16° </s>
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98 </p>
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99 <p type="main">
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100 <s>Nelle Meccaniche. Nel cap. 17° </s>
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101 </p>
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102 <p type="main">
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103 <s>Nell’Architettura ornata e militare. Nel cap. 18° </s>
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104 </p>
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105 <p type="main">
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106 <s>Nell’Arte militare. Nel cap. 19° </s>
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107 </p>
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108 <p type="main">
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109 <s>Nell’Agricoltura. Nel cap. 20° </s>
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110 </p>
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111 <p type="main">
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112 <s>Nella Navigatoria. Nel cap. 21° </s>
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113 </p>
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114 <p type="main">
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115 <s>Nel Disegno, in quanto abbrevia la Pittura, la Scoltura e la Plastica. Nel
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116 cap. 22° </s>
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117 </p>
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118 <p type="main">
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119 <s>Nell’Arti fabrili. Nel cap. 23° </s>
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120 </p>
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121 <p type="main">
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122 <s>Se alcuna volta l’angolo si può chiamar tagliamento di linea, di figura o di
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123 corpo. Nel cap. 24° </s>
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124 </p>
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125 <p type="main">
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126 <s>Se gli angoli solidi e piani sieno i medesimi. Nel cap. 25° </s>
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127 </p>
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128 <p type="main">
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129 <s>Tutte queste cose prenderemo a considerare per dimostrar altrui in qual cosa
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130 consista la natura, la proprietà e la vista dell’Angolo, il quale via più
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131 prontamente d’ogni altra cosa geometrica ci si è offerto perciò che, essendo
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132 due gli accidenti della linea, principale elemento della geometria, uno che
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133 è l’esser infinita, e potersi dividere e terminare: e l’altro il cagionar
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134 col mezzo del toccamento d’altra linea l’angolo, ed esendo più noto
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135 quest’effetto che altro accidente, questo solo ci dà occasione di
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136 contemplare tutto quello che appartiene alla sua natura. </s>
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137 <s>Onde con l’aiuto del divino favore daremo cominciamento.</s>
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138 </p>
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139 <p type="main">
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140 <s>//</s>
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141 </p>
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142 </chap>
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143 <chap>
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144 <p type="head">
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145 <s>Se l’angolo sia cosa reale o intelligibile o immaginaria</s>
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146 </p>
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147 <p type="head">
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148 <s>Cap. 1</s>
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149 </p>
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150 <p type="main">
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151 <s>In due maniere al parer mio par che l’angolo si possa considerare, cioè o in
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152 quanto al fondamento e alla materia in cui si truova (che non è
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153 inconveniente far questa consideratione nella Geometria; perciochè la
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154 Matematica considera le cose in quanto sono in materia, benchè di essa
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155 niente si curi!) overo in quanto a se stesso, cioè come è fatto di cose
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156 intelligibili, ed in quanto la sua forma è più tosto intelligibile che
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157 sensibile; che non è altro che un dire che degli Angoli altri sono
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158 materiali, altri intelligibili, i quali sono puramente matematici. </s>
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159 <s>E per cominciar da questo membro della divisione con ordine contraposto dirò
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160 che sì come dalla materia sensibile si ritrahe la forma sensibile (almeno
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161 per accidente se non per sé) così per una certa proportione dalla potenza
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162 della materia intelligibile si cava la forma intellligibile. </s>
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163 <s>Perciochè se le Matematiche hanno la materia intelligibile, che in quanto
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164 materia non è senza potenza overo habilità: e questa non dee esser
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165 superflua; dunque necessariamente bisogna che da essa si tragga qualche
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166 forma, la quale non può esser non proportionata alla materia; però se la
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167 materia è intelligibile, la forma sarà intelligibile. </s>
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168 <s>La materia dell’angolo senza dubbio alcuno è ‘l punto e le linee, che non
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169 sono obbietti del senso per loro stessi, ma dello ‘ntelletto, e perciò si
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170 appellano intelligibili; onde ragionevolmente la materia dell’Angolo è
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171 intelligibile, dalla quale è inconveniente farsi astrattione. </s>
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172 <s>La forma di esso, che si toglie dal grembo di questa materia è lo stesso
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173 contatto delle linee in uno stesso punto, overo la stessa inclinatione e
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174 congiognimento in un punto comune. </s>
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175 <s>La quale è intelligibile perciochè solamente si apprende <pb pagenum="folios 2v-3r"></pb>dallo intelletto e perché prende la qualità
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176 dalla natura della materia onde si leva. </s>
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177 <s>Oltre acciò se quella forma si appella materiale; che si stacca dal seno
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178 della materia e quella immateriale che da essa non si ritrahe; per qual
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179 cagione quella forma che si cava dalla potenza della materia intelligibile
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180 non si potrà dire intelligibile? Adunque, per tutte queste ragioni
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181 concludiamo l’Angolo esser cosa intelligibile. </s>
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182 <s>Ma si cerchi hora se è vero che sia cosa reale, cioè in quanto al fondamento
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183 ed alla materia da cui si regge. </s>
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184 <s>Chiara cosa è che l’Angolo, fra tutta l’università delle cose, non si può
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185 truovar se non in due generi, cioè o nel naturale, o nell’artificiale;
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186 perciochè se riguardiamo le cose naturali, o l’artificiali vi scorgeremo
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187 ogni maniera d’angoli. </s>
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188 <s>Perciochè le cose artificiali, o sieno opere overo stromenti, quasi tutte son
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189 terminate con figure, che hanno qualche specie d’angolo, escettuate però
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190 tutte quelle che hanno la figura sferica e ritonda. </s>
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191 <s>Se altri volesse addurre gli essempij di tutte si allongarebbe troppo il
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192 ragionamento, anzi sarebbe superfluo, essendo tutto ciò notissimo a
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193 ciascuno. </s>
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194 <s>Questo solo dirò degli Angoli artificiali, che essendo essi o a squadra o a
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195 soprasquadra, o a sottosquadra, sì come si ritrahe da Leon Battista Alberti
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196 nell’Architettura, non posson non esser reali; perciochè vengono così
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197 appellati da uno stromento reale, e naturale, che gli produce, il quale è la
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198 squadra, che materialmente mostra l’angolo retto. </s>
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199 <s>E le cose naturali essendo per lo più corpi terminati da qualche ragion di
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200 figura, oltre a quelli che son di figura in tutto d’Angoli nuda, sono in
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201 gran parte angolari; ciò sono o pietre o gemme o piante overo isole. </s>
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202 <s>Fra l’isole due sono nell’Italia nel lago dell’Anguillara, che si mostrano
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203 hora di figura triangolare hora di ritonda; ma di quadrata <lb></lb>//<lb></lb>non
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204 mai, sì come si può vedere appo Giorgio Agricola nel 2° lib. </s>
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205 <s>Della natura delle cose, che dalla terra scorreno. </s>
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206 <s>Le gemme si truovano di figure angolari, come l’Androdomante, il Basalte
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207 Miseno, il Cristallo, il Pangonio, il Diamante ed altre come si può vedere
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208 appo Giorgio Agricola nel primo lib. </s>
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209 <s>Della natura delle cose fossili. </s>
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210 <s>Così ancho le pietre si trovano angolari, come sono quelle presso a Bolseno
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211 in una grotta dove mostrano quasi una muraglia simile a quelle degli antichi
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212 Romani, d’opera reticolata. </s>
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213 <s>Si truovano ancora alcune pietre sciolte di grandezza alquanto meno di una
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214 nicciuola e di colore simile al ferro di figura esagona formate a ponta di
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215 diamante da ogni banda in guisa tale che mostrano due piramidi esagone. </s>
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216 <s>Nella margassita si truovano alcune volte noccioletti di forma cubica; e così
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217 ancho fra queste cose si veggano altre figure angolari. </s>
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218 <s>Le piante anchora si mostrano la maggior parte angolari; perciochè il Cipero,
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219 il Gionco odorato, l’Assaro, la Cicerbita, l’Aragallide, la Centaurea
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220 minore, la Baccara, il Marzobbio, l’Hormino, la Bettonica, il Sinfito, il
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221 Climeno, l’Otiopide, il Cirsio, il Bunio, la Lappa minore ed altre anchora
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222 hanno diversamente i fusti angolari. </s>
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223 <s>Oltre acciò i fiori, le foglie e ‘semi sono in diverse maniere angolari, come
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224 ‘l seme del Sezeli Massiliense è quadrato, sì come si può ritrarre da
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225 Dioscoride. </s>
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226 <s>Di modo che quindi possiamo concludere che per cagion delle cose naturali, ed
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227 artificiali, nelle quali vediamo le specie degli Angoli, l’Angolo habbia
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228 qualche realità. Oltre a questo quelle cose sono reali, che si ritruovano
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229 fuor dello ‘ntelletto humano; onde l’Angolo essendo nelle cose artificiali o
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230 nelle naturali si truovarà insieme con esse fuor dello ‘nteletto e perciò
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231 potrà dirsi reale. </s>
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232 <s>Ansi, queste fuor dello ‘ntelletto <pb pagenum="folios 3v-4r"></pb>non possono
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233 esser se non corporee terminate da qualche maniera di figura: overo
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234 appartenenti a corpi in quanto son termine di essi, o delle superficie loro;
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235 onde necessariamente bisogna che habbiano Angoli e sotto essi apparischino. </s>
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236 <s>Perciochè (sì come il Padre Ignatio Danti nella Prospettiva d’Euclide, sopra
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237 la 8a suppositione di esse) tutte le cose visibili vedersi sotto qualche
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238 angolo, poiché la figura compresa da’ raggi visuali è un conio la cui ponta
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239 si ferma nel centro dell’occhio, nella quale i medesimi raggi formano angoli
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240 diversi secondo la diversità delle cose vedute. </s>
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241 <s>Onde sì come l’atto del vedere de’ nostri occhij è cosa reale, così ‘l conio
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242 formato da’ raggi visuali sarà reale; perciochè il vedere stesso non si
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243 forma se non col mezzo del detto conio, la cui base è l’obbietto e ‘l cui
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244 termine è ‘l centro dell’humor cristallino dell’occhio, amendue cose reali;
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245 di maniera che essendo reale il conio non può non esser reale l’angolo da
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246 esso formato. </s>
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247 <s>Non starò hora a cercar se la visione si faccia per ricevimento di specie,
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248 come vogliono i Peripatetici: o per mandar fuore i raggi visuali, come piace
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249 a’ Platonici. perciochè questo luogo non è proportionato a tali questioni;
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250 onde le lassarò trattare a’ Filosofi, mentre delle cose dell’anima
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251 discorrono. </s>
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252 <s>Mi bastarà bene haverne accennato in quanto a quella parte che
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253 matematicamente si può considerare per pruovar la realità dell’Angolo. </s>
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254 <s>Da queste cose anchora si può ritrarre che l’Angolo sia cosa reale; che si
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255 ritruova fra i dieci generi delle cose detti da’ Filosofi “Predicamenti”.
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256 Onde Proclo sopra Euclide mostra che de’ Filosofi antichi altri posero
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257 l’angolo nel predicamento della relatione altri nella quantità, ed altri
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258 nella qualità; in qual <lb></lb>//<lb></lb>unque sia di questi generi, e cagione che
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259 inferiamo che sia cosa reale. </s>
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260 <s>Ma vediamo hora se l’Angolo si possa dir cosa immaginaria. </s>
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261 <s>Le cose immaginarie sono di due ragioni; perciochè o sono puramente
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262 immaginarie o impuramente. </s>
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263 <s>Le impuramente immaginarie son quelle che parte son reali e parte immaginate. </s>
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264 <s>La parte reale si è ‘l primo concetto che è fondamento e origine della
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265 immaginatione, cioè del secondo concetto: overo quella simiglianza che ha la
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266 immaginatione della cosa con la cosa reale, o con quella parte che è reale. </s>
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267 <s>Perciochè (come dice Temistio nel 3° dell’Anima, cap. 13°) l’immaginatione è
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268 un certo vestigio del senso; perciochè si forma dalle cose sensate. </s>
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269 <s>E però la cosa immaginaria, non puramente tale è un certo vestigio della cosa
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270 sensata. </s>
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271 <s>Ma la parte immaginaria si è quella cosa che è similitudine del secondo
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272 concetto separato in tutto dalla cosa reale. </s>
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273 <s>Le puramente immaginarie son quelle cose, che l’Immaginativa per se stessa si
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274 è fabbricate senza haverne alcuna simiglianza nelle cose sensate, e per
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275 dirla più apertamente sono le stesse fintioni e le stesse chimere e capricci
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276 che fuor della Immaginativa non hanno essere. </s>
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277 <s>Però Alesandro Afrodiseo sopra ‘l primo libro dell’Anima, nel cap. 2°, divise
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278 le immaginationi in vere e false. </s>
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279 <s>Quasi dir volesse che le pure immaginationi fosser false e le non pure vere. </s>
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280 <s>Le pure immaginationi, per Gio. </s>
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281 <s>Grammatico, nel 3° dell’Anima, parte 16a, son quelle che fingono le forme: e
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282 le non pure quelle che ricevono le forme, cioè le specie e simiglianze dalle
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283 cose sensate e reali. </s>
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284 <s>Così le cose immaginarie, che sono effetti dell’immaginatione, saranno o vere
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285 o false, o finte o reali, o pura fintione, o simiglianza di cosa sensata e
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286 reale. </s>
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287 <s>Hora per applicar tutto questo discorso al proposito nostro diremo <pb pagenum="folios 4v-5r"></pb>che l’angolo considerato, in quanto si fonda in
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288 cosa reale, essendo uno effetto delle linee ritruovate in materia, non dee
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289 dirsi puramente cosa immaginaria, ma non puramente, cioè in quanto alla
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290 dependenza che è schiettamente sensata e reale: ed in quanto alla
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291 simiglianza d’una cosa reale. </s>
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292 <s>E così si può dire, che l’Angolo sia cosa immaginaria, ma nel secondo modo
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293 non si conviene affermare; perciochè sarebbe al tutto cosa falsa, e le
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294 Matematiche non sarebbero fondate in principij veri e certi; ma in falsi ed
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295 incerti. </s>
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296 <s>Che tanto è dire l’Angolo esser puramente immaginario quanto falso e incerto:
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297 e ponendosi falso e incerto, le linee e ‘l punto che concorrono alla sua
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298 produttione sarebbero al tutto falsi ed incerti, onde seguirebbe la
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299 Geometria esser collocata in cosa non vera e non haver certezza alcuna. </s>
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300 <s>La qual cosa fu ingegnosamente avvertita da Niccolò Tartaglia nel primo lib.
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301 della sua Geometria, nel cap. primo. </s>
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302 <s>Adunque o non bisogna dire che l’angolo sia cosa immaginaria o se pur si dice
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303 più tosto si affermi che sia cosa non puramente immaginaria, che altramente;
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304 perciochè in questa maniera è più ragionevole.</s>
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305 </p>
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306 <p type="main">
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307 <s></s>
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308 </p>
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309 </chap>
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310 <chap>
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311 <p type="head">
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312 <s>Di quante maniere sia l’angolo</s>
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313 </p>
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314 <p type="head">
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315 <s>Cap. 2</s>
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316 </p>
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317 <p type="main">
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318 <s>Fra le cose naturali ed artificiali due generi sono che abbracciano la
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319 diversità degli Angoli. </s>
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320 <s>Il primo contiene tutte le superficie piane, il secondo tutti i corpi solidi,
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321 perciochè sì come al producimento degli Angoli nelle superficie piane
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322 concorreno le linee così alla costitution di essi ne’ solidi concorreno
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323 insieme con le linee le superficie anchora; perciochè sì come nel primo modo
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324 le linee son fra loro inclinate, così nel secondo sono le superficie, come
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325 si può trarre dalla quinta definitione dell’undecimo libro di Euclide. </s>
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326 <s>E questa divisione si ritrahe da Proclo. </s>
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327 <s>Ma ripigliando la di <lb></lb>//<lb></lb>visione, si dee ripartire tutto ‘l genere
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328 degli Angoli, che si ritruovano nelle superficie piane; secondo la qualità
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329 delle figure che le terminano; perciochè altre sono di figura rettilinea,
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330 altre di curvilinea, e altre di mista; di modo che gli Angoli delle
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331 rettilinee si dicano rettilinei, que’ delle curvilinee curvilinei, que’
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332 delle miste misti: e come le due prime specie di figure sono simplici, così
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333 gli angoli loro sono simplici: e come la terza specie delle figure è mista,
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334 così gli angoli son misti. </s>
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335 <s>Così anchora considerando di nuovo il 2° genere si vede tutta la diversità
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336 degli Angoli de’ corpi solidi nascer dal vario producimento loro. </s>
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337 <s>Perciochè o si producano dal tagliamento fatto da diversi cerchi, o dal
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338 diametro, o dall’asse, come avviene nella sfera, ansi in ogni corpo sferico:
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339 o dal percotimento, che fanno i raggi del Sole nello specchio concavo, o
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340 convesso, mentre si ripiegano: o dal cadimento, che fanno le cose sopra ‘l
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341 piano, le quali balsando si torrono ‘l movimento, come le palle da giuocare
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342 battute sopra ‘l piano: o dal concorso delle superficie del corpo solido
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343 ugualmente e da figure uguali e della medesima specie terminato e da linee
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344 terminanti uguali, come avvien ne’ corpi regolari, overo da superficie
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345 variamente figurate e disuguali e di specie diversa terminato disugualmente
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346 insieme con linee disuguali, come accade ne’ corpi irregolari: o da linee,
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347 che habbiano diversa positione, come se altre sieno nel piano, ed altre
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348 sopra ‘l piano cadenti sopra esse, in maniera che in uno stesso tempo le
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349 tocchi tutte in guisa tale che d’ogni ‘ntorno produca gli angoli uguali,
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350 come avviene negli spatij e ne’ vani delle stanze: o da due superficie piane
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351 circolari opposte, e da una curva che col mezzo di due circonferen <pb pagenum="folios 5v-6r"></pb>ze si congiogne con essa ad angoli retti, come
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352 avviene nel cilindro o nella colonna: o da due superficie una tonda e
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353 conessa ristretta in un punto ed una piana e circolare, che è la base, che
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354 seco si congiogne per una circonferenza, come si vede nel conio: overo dal
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355 raccoglimento di cinque superficie, che per mezzo di linee disuguali si
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356 legano insieme delle quali la minore che è base è opposta all’Angolo della
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357 cima, ed è di figura quadrata; e questo avviene alla piramide. </s>
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358 <s>Di maniera che ne’ solidi quante sono le maniere del producimento degli
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359 angoli, tanti sono gli angoli; in quanto che ‘l diverso modo gli dà qualche
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360 diversità, come si vede chiaro da chi bene osserva el componimento di tutti
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361 i corpi solidi. </s>
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362 <s>Ma finalmente tutte le specie degli angoli solidi si riducano a due generi,
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363 cioè all’angolo composto ed all’angolo simplice: al composto: al composto
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364 che si mostra in due foggie, nella prima cioè come è composto d’angoli di
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365 diversa specie, come di rettilineo e misto: nella seconda come è composto
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366 d’angoli della medesima specie, cioè di rettilinei: al simplice in quanto
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367 che si riduce all’angolo minore che è elemento degli altri angoli e questo
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368 ne’ solidi è composto di più angoli minimi, ed è simplice rispetto agli
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369 altri angoli massimi de’ solidi, poiché questi son composti di quelli, come
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370 si vede nell’acutezza delle piramidi. </s>
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371 <s>E per abbracciar insieme tutti gli angoli e piani e solidi, bisogna insieme
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372 con Euclide ridurgli a tre specie, ciò sono retto, ottuso ed acuto.</s>
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373 </p>
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374 <p type="main">
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375 <s></s>
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376 </p>
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377 </chap>
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378 <chap>
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379 <p type="head">
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380 <s>Che cosa sia l’angolo ed in che sia collocata la sua essenza</s>
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381 </p>
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382 <p type="head">
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383 <s>Cap. 3</s>
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384 </p>
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385 <p type="main">
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386 <s>E’ cosa convenevole che, avanti che ricerchiamo la definition dell’angolo,
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387 vediamo il significato del suo <lb></lb>//<lb></lb>nome. </s>
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388 <s>E adunque l’angolo detto da’ Greci “gonia”; onde le cose angolari grecamente
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389 si appellano “goniadis”; onde Varrone, nel primo libro della Lingua latina
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390 dice che l’angolo però si appella gonia; perciochè in esso sia un luogo
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391 angustissimo; che non significa altro che un certo restrignimento, che
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392 risulta da due linee, che si terminano in un ponto commune; poiché non per
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393 altro da Proclo è detto una simiglianza di ristregnimento. </s>
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394 <s>Oltre acciò l’angolo non denota altro, che un certo gombito, o cantone,
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395 grecamente detto anchon, prodotto dal piegamento e congiognimento di due
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396 righe, sì come si può vedere appo Vitruvio nel lib. 3° cap. 3°, dove
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397 favellando delle colonne scannellate, dice la scannellatura farsi in guisa,
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398 che postavi dentro la squadra, e girata intorno, tocchi con le sue piegate
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399 righe il concavo e l’estremità del canale. </s>
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400 <s>Il piegamento adunque che fa la squadra formando angolo retto si dice anchon
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401 grecamente; perciochè le righe che la compongono separate dalla squadra
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402 hanno altro nome; che si appellano regole; ma nella squadra, perché
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403 congiognendosi formano un piegamento a modo di gombito che non è altro che
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404 ‘l luogo, dove si congiongano l’ossa del braccio, che si chiama cubito,
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405 overo gombito. </s>
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406 <s>Però non più regole, ma anchone si chiamano; ansi i piegamenti di queste
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407 righe in fra loro scambievoli sono chiamati anchones da Guglielmo Filandro,
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408 nell’ Annotationi sopra Vitruvio nel 3° lib. 3° cap., sì come anchora da
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409 Vitru. nel 8° lib. cap. . . si esplica. </s>
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410 <s>Perciochè quivi egli dice que’ piegamenti, che fanno angoli retti si chiamano
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411 anchones, sì come vediamo nella squadra, etc. </s>
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412 <s>E molto meglio Daniel <pb pagenum="folios 6v-7r"></pb>Barbaro l’esplicò;
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413 perciochè egli esponendo ‘l medesimo luogo, disse: anchones sono le braccia
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414 della squadra etc. </s>
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415 <s>Dalla simiglianza forse di questo piegamento, Ancona, città della Marca, è
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416 stata così nominata; per cagion della figura, che rappresenta il suo posto. </s>
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417 <s>Così anchora è nominata una parte del Nilo, la quale è navigabile finalmente
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418 agone, overo agonia è un luogo, dove si facevano i giuochi degli antichi,
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419 nel quale, nella parte principale, era di figura non angolare; che perciò
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420 (secondo che è piaciuto ad alcuni) era chiamato agone, cioè privo d’angoli. </s>
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421 <s>Oltre acciò, se si riguarda bene l’angolo, non è altro che quell’accostamento
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422 di due cose vicine, le quali, da diverse bande tirate, si strengano per far
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423 la forma dell’angolo: e vien dalla parola englis (sì come dice Festo) che
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424 vuol dir “esser” vicino ciò che si ha da ristregnere; che egli dice
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425 “englis”, cioè che presso si accosti. </s>
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426 <s>Ma lassiamo da parte queste cose le quali in qualche parte si allontanano
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427 dall’essenza dell’angolo; che in questo luogo intendiamo cercare di
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428 definirlo con una definitione, o almeno con una descrittion commune. </s>
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429 <s>Dico adunque cercar l’essenza dell’angolo non vi esser altro mezzo che
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430 l’osservation del suo producimento. </s>
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431 <s>Noi vediamo formarsi l’angolo con la inclinatione delle linee ed eseguirsi
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432 diversamente, cioè o dirittamente overo obliquamente; obliquamente quando da
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433 tali inclinationi risultano angoli disuguali: dirittamente quando ne
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434 risultano angoli uguali. </s>
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435 <s>Dirittamente dico, non quando una linea è per diritto dell’altra, o si
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436 continua con l’altra; <lb></lb>//<lb></lb>ma quando sopra una linea cade un’altra
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437 perpendicolare. </s>
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438 <s>Obliquamente, quando una linea si piega o più o meno sopra la linea piana;
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439 onde più o meno si accosta alla linea piana. </s>
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440 <s>Supposta questa cognitione, facilmente potremo addurre una deffinition
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441 commune a tutti gli angoli ed è:</s>
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442 </p>
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443 <p type="main">
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444 <s>“L’angolo è un concorso e contatto di linee in un punto commune; che per la
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445 diritta e per l’obliqua inclinatione di esse, riceve diversità di specie e
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446 di grandezze”.</s>
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447 </p>
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448 <p type="main">
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449 <s>E’ un concorso di linee perciochè tutte le linee, che fanno angolo concorreno
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450 insieme, cioè convengono e si congiogano in un punto, il che non avviene
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451 alle parallele, come è manifesto per la trentacinquesima def. del primo d’
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452 Euclide; che se accadesse, non sarebbero più parallele, come si vede nel 5°
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453 postulato e nella dimostration del 18° Teorema del primo. </s>
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454 <s>Perciochè esse non fanno angolo; onde avviene che non chiudino spatio da
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455 banda alcuna. </s>
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456 <s>E’ un contatto; perciochè nel concorrere insieme le linee si toccano in uno
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457 stesso ponto, che le congiogne, e continua: e questo concorso e contatto è
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458 commune a tutte le maniere degli angoli, o rettilinei o curvilinei, o misti. </s>
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459 <s>Si dice farsi in un ponto; perciochè le linee che producono e chiudono
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460 l’angolo, finiscono in un ponto commune, che è termine, e congiognimento
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461 delle linee, che si toccano insieme. </s>
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462 <s>Onde quantunque per loro stesse sieno contigue, con tutto ciò, col mezzo del
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463 punto divengon continue; sì come insegna Aristotile nel lib. … della
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464 Filosofia naturale part. … Ma per intender questo si dee notare, che altro è
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465 continuarsi simplicemente, che è lo stesso che allongarsi infinitamente, o
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466 indeterminatamente, come si vede nella dimostratione del Teor. 9° del primo
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467 d’Euclide <pb pagenum="folios 7v-8r"></pb>e ‘n quella del Teor. 13° e del Prob.
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468 10, ed altro è continuarsi col mezzo del ponto, quasi che esso sia ‘l
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469 collegamento e ‘l chiodo, che congiogne due linee disgionte. </s>
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470 <s>Nel primo modo s’interpon sì bene ‘l ponto nel continuamento della linea, ma
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471 continuata si perde; perciochè non vi si cagiona l’angolo, o però non
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472 bisogna che vi rimanga ‘l ponto. </s>
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473 <s>Nel secondo in maniera s’interpone che vi rimane; perché mentre è mezzo e
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474 legamento delle linee contigue, è anchora termine commune di esse, e
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475 terminamento dell’angolo. </s>
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476 <s>Nella stessa guisa si può formar la definitione commune dell’angolo solido,
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477 ritrahendola dall’osservatione del suo nascimento. </s>
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478 <s>Ma solamente vi sarà diversità in quanto che ‘l concorso e ‘l contatto si fa
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479 non di linee sole, ma di linee e di superficie insieme: non terminante in un
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480 ponto solo, ma ancho in una linea commune, che riceve diversità di specie e
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481 di grandezze e per la diversità delle inclinationi di esse. </s>
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482 <s>A questo proposito non mi par disdicevole, per maggior dichiaratione
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483 dell’angolo solido, notar la definitione formata da Bonaiuto Lorini nella
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484 sua Fortificatione; perciochè egli nella quarta definitione dice: l’angolo
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485 si addomanda quella parte, dove due linee si congiongano insieme, cioè AB.
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486 BC., che si congiongano in B. dove formano in tal parte l’angolo. </s>
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487 <s>Dice quella parte perché intende dell’angolo materiale e solido; perciochè,
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488 nella fortificatione, come in tutta l’architettura, non solamente i ponti,
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489 le linee e le superficie e ‘corpi si considerano come applicati alla materia
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490 sensibile e trattabile, così ancho gli angoli che in tali cose consistono.
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491 </s>
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492 </p>
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493 <p type="main">
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494 <s>Quindi adunque non solamente è manifesto qual sia l’essenza dell’angolo, ma
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495 anchora in qual cosa consista, cioè nella materia intelligibile, che sono le
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496 linee e le superficie, volendo noi non trapassare i confini della pura
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497 Geometria: e nella forma parimente intelligibile, che è l’accostamento e
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498 contatto di due linee condotto e terminato in un ponto: overo di tre o più
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499 superficie, terminato in un ponto ed in una o più linee, come si vede ne’
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500 cinque corpi regolari.</s>
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501 </p>
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502 <p type="main">
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503 <s></s>
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504 </p>
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505 </chap>
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506 <chap>
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507 <p type="head">
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508 <s>Che per la divisione del cerchio si ritruovano tutte le specie degli
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509 angoli</s>
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510 </p>
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511 <p type="head">
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512 <s>Cap. 4</s>
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513 </p>
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514 <p type="main">
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515 <s>Per intender meglio questa propositione, fa di mestiero supporre, come cosa
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516 nota e pruovata per la <lb></lb>//<lb></lb>Geometria la verità d’un’altra
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517 propositione così fatta. </s>
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518 <s>Ogni angolo potersi formare con la scambievole intersegation delle linee
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519 rette o curve. </s>
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520 <s>E posta già questa notitia ed essendo vero che ogni figura angolare si forma
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521 dalla varia division del cerchio e non possendosi formar figura alcuna,
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522 senza la varia costitution degli angoli, sarà anchora vero che per la
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523 diversa division del cerchio si truovino tutte le specie degli angoli; onde
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524 avendo noi primieramente cognitione e certezza della verità del primo
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525 supposto, veniamo facilmente in cognitione e certezza della verità del
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526 secondo. </s>
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527 <s>E per confermation del primo si dee considerare, che le figure o regolari o
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528 irregolari son di due maniere, altre al tutto dependenti da altra figura
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529 simplice, uniforme e perfetta: altra non dependente da alcuna; ma da essa
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530 scaturiscono tutte. </s>
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531 <s>Quelle di questa fatta sono tutte le figure circolari; perciochè ‘l cerchio è
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532 quasi materia di tutte le figure rettilinee; poiché dal diverso tagliamento
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533 di lui si traggano tutte le specie di figure rettilinee. </s>
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534 <s>Ma quelle dell’altra maniera sono tutte le figure rettilinee che dal detto
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535 tagliamento si formano, e perciò dependono dalla figura uniforme, simplice e
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536 perfetta, che è la circolare, la quale non ha dependenza da altra figura;
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537 che perciò si è detta simplice, ma da essa dependono tutte; poiché da lei si
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538 traggano, come dal seno della materia, col mezzo del taglio diversamente
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539 fatto. </s>
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540 <s>Adunque tagliato variamente ‘l cerchio si formano diverse foggie di
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541 rettilinee figure, e così per conseguenza diverse maniere d’angoli. </s>
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542 <s>Ma per intender meglio tutta la natura del detto tagliamento, si dee
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543 osservare, che in diversi modi dalla pratica geometrica si può eseguire;
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544 perciochè o si fa dalla metà del diametro o dal diametro intero, o dalla
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545 diversa division del diametro o dalla corda o dalla linea detta saetta, che
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546 (secondo <pb pagenum="folios 8v-9r"></pb>Daniel Barbaro, nel primo lib. di
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547 Vitru. cap. 5) è quella linea, che dal mezzo della corda con angoli uguali
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548 ascende alla circonferenza dell’arco: o dalla intersegation fatta da altri
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549 cerchi. </s>
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550 <s>Consideriamo adunque ciascuna specie di questi tagliamenti del cerchio, acciò
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551 con più facilità veniamo a ritrarne il nascimento di qualunque maniera
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552 d’angolo, e primeramente con ordine ripiegato, si consideri come dalla
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553 intersegation de’ centri si producano gli angoli. </s>
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554 <s>Questo si dimostra da Euclide nel primo Problema del primo degli Elementi, là
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555 dove dalla intersegation di due cerchi trahe la formation del triangolo
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556 equilatero; onde risultano tre angoli acuti. </s>
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557 <s>Nel qual luogo il Commandino, con la medesima intersegation de’ cerchi, ma
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558 fatta diversamente, formando i triangoli equicruri e scaleni produce gli
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559 angoli acuti e gli ottusi: e noi anchora variando ‘l modo della
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560 intersegatione nella sposition del medesimo Probl. formando altro triangolo
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561 ritroviamo l’angolo retto, come si vede qui appresso. </s>
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562 <s>Nel qual modo si suppon la pratica insegnata da Euclide e si allonga da ogni
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563 banda la linea retta AB. e posto ‘l centro D. e lo intervallo DE. si
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564 descriva ‘l cerchio EFG. così posto ‘l centro E. e lo in ED si descriva ‘l
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565 cerchio DFH. Di maniera che amendue si taglino nel punto F. Quindi l’arco
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566 FD. si divida in tre parti uguali una delle quali si trasporti due volte
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567 nell’arco FH., cioè accanto al taglio F. che saranno nei tagli FKL e divisa
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568 la parte KL. per mezzo, vi si faccia ‘l taglio I. che sarà per diritto sopra
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569 ‘l centro E. Di poi si congiongano i tagli ID. IE. vedremo cagionarsi ‘l
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570 triangolo rettangolo IED. <lb></lb>//<lb></lb>con l’angolo retto IED. costituito
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571 dalla perpendicolare IE. per la decima def. del primo d’Euclide. </s>
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572 <s>Alla qual cosa si sottoscrive Amonio sopra Porfirio dicendo, che gli angoli
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573 retti non posson muoversi se non al centro. </s>
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574 <s>Perciò che quivi consiste naturalmente la quiete; onde da Aristotile nelle
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575 Meccaniche, tale angolo è detto angolo della quiete. </s>
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576 <s>E ‘l punto E. dove detto angolo si termina non è altro che ‘l centro del
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577 cerchio DFH. e ciò avviene per la intersegation de’ cerchi minori e maggiori
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578 col mezzo de’ quali si truova ‘l punto onde dee partirsi la perpendicolare e
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579 ‘l punto dove dee terminare sopra una linea retta piana già data la quale è
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580 DBAC. e che ciò sia vero, con la stessa operatione del su detto primo Prob.
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581 facendo centro il segno D. e intervallo la linea DH. si faccia la portion
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582 del cerchio LM. e facendo centro ‘l segno H e intervallo HD. si faccia la
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583 portione del cerchio KN. la quale tagliarà la LM. nel segno O.
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584 necessariamente la perpendicolare che da esso si muovarà cadrà nel centro;
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585 perciochè ‘l taglio O. e ‘l centro E. son diritto fra loro, e si fanno
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586 termini della linea perpendicolare. </s>
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587 <s>Oltre acciò, pel secondo Postulato di Euclide, allongata la perpendicolare
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588 IE. fine al P. che finisca nella circonferenza del cerchio DFH. e allongata
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589 la linea piana DE. fine in H. che termini nella circonferenza del medesimo
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590 cerchio necessariamente il detto cerchio verrà tagliato in quattro quarte
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591 uguali, perciochè le linee DH. IP. sono diametri che lo dividono prima in
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592 due e poi in quattro parti uguali; che la DH. il taglia per metà e la IP.
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593 divide la detta metà in due parti uguali, e questo avviene infallibilmente
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594 perciochè i diametri che dividono due volte ‘l cerchio si tagliano fra loro
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595 in due parti uguali; perciochè passano pel centro; che se non passassen pel
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596 centro non potrebbero dividersi insieme in due parti uguali come si dimostra
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597 da Euclide nella quarta prop. del 3°, onde non si taglierebbe ‘l cerchio in
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598 quattro quarte uguali; determinate da quattro <pb pagenum="folios 9v-10r"></pb>tagliamenti ugualmente lontani fra loro, cioè DIHP. i quali congionti con
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599 quattro linee si costituiscan quadrato e per conseguenza quattro angoli
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600 retti; perciochè ciascuno è nel suo mezzo cerchio, come si può pruovare con
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601 la trentunesima del 3° e ciascun lato corrisponde a ciascuno arco della
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602 quarta del cerchio; che dalla quarta parte del cerchio si trahe la quarta
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603 del quadrato; che tal proportione ha un lato del quadrato con la parte dello
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604 spatio a lui sottoposta a tutto ‘l quadrato, quale ha una quarta del cerchio
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605 a tutto ‘l cerchio. </s>
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606 <s>Si costituiranno dico da quattro tagliamenti del cerchio quattro angoli retti
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607 che non possano non essere uguali fra loro sì perché questa è proprietà
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608 degli angoli retti; sì perché ne’ detti tagliamenti si costituisce lo spatio
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609 parallelogrammo, che ha gli angoli e lati opposti uguali fra loro, come si
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610 vede nella prop. trentaquattresima del primo. </s>
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611 <s>Onde si potrebbe ritrar un corollario, cioè:</s>
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612 </p>
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613 <p type="main">
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614 <s>“Che dentro un cerchio, ritruovate quattro quarte uguali e stabilito un
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615 triangolo rettangolo si costituisca un quadrato e truovato pel suo
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616 tagliamento un angolo retto, facilmente si possa haver cognition di quattro
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617 retti.”</s>
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618 </p>
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619 <p type="main">
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620 <s>Ma questo, oltre che è noto per le cose dette, non si dimostrarà per fuggir
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621 la longhezza. </s>
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622 <s>Si è detto che dalla saetta si divide ‘l cerchio onde si cagionano gli
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623 angoli. </s>
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624 <s>Perciò che questa linea che è perpendicolare in quanto all’effetto, ma è
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625 detta metaforicamente saetta, sorgendo dal mezzo della corda, non può non
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626 tagliare il cerchio; e questo accadendo diversamente secondo la diversa
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627 positione della corda, ancho ‘l cerchio diversamente vien diviso; onde
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628 quantunque da questa linea che è retta sia tagliato il cerchio ad angoli
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629 retti sferali, od almeno da essa, e dalla circonferenza sieno costituiti
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630 angoli misti; con tutto ciò considerata la dispositione e l’habilità de’
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631 tagliamenti fatti nel cerchio, vedremo espressamente sorger
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632 <lb></lb>//<lb></lb>diverse specie d’angoli, secondo la diversità delle figure
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633 rettilinee che per essi tagli si formano, come vedrà con esperienza
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634 qualunque è esercitato nelle pratiche del Compasso nel formar le figure
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635 moltilatere, con la duplication de’ lati ritruovata dal taglio fatto per
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636 modo di saetta con la forza dell’arcuatione, come in altro luogo da noi s’è
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637 ragionato alla longa, e come si vede in questa figura. </s>
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638 <s>La qual cosa si vede espressa nella trentunesima del 3°, là dove si mostra
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639 l’angolo minore del retto esser nella maggior portione del cerchio, e
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640 l’angolo maggiore esser nella minore e quindi nasce la diversità delle
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641 figure e degli angoli, cioè dalla diversità delle portioni de’ cerchi sotto
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642 le quali si truovano.</s>
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643 </p>
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644 <p type="main">
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645 <s>Si è detto ‘l cerchio dividersi dalla corda, la quale è quella linea retta
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646 che ‘l taglia non passando pel pel centro in due portioni disuguali per la
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647 diciannovesima def. del primo degli Elementi. </s>
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648 <s>Si appella corda dagli Architetti e dagli artefici, perché si servano delle
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649 cose geometriche adattandole alla materia. </s>
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650 <s>Questa linea tagliando ‘l cerchio è cagione degli angoli retti e degli acuti,
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651 mentre sopra i tagliamenti fatti nella sua circonferenza si formano i
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652 quadrati e triangoli equilateri, come si vede qui nell’essempio. </s>
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653 <s>Ciascun lato di queste figure si appella corda perciochè taglia ‘l cerchio in
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654 due portioni minori e maggiori AB e CDAB. overo ACBD e DB. nel primo
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655 essempio DEF. ED. nel secondo. </s>
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656 <s>Si divide anchora ‘l cerchio per la diversa division del diametro. </s>
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657 <s>Cioè dividendo prima ‘l diametro in vintiquattro particelle uguali, e per
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658 formare ‘l triangolo equilatero, acciò vediamo gli angoli acuti, <pb pagenum="folios 10v-11r"></pb>bisogna porre un piè del compasso in una
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659 estremità del diametro, allargando l’altro tanto che prenda la vintuna
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660 particella, e con tal misura facendo l’arco, si tagli da ogni banda la
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661 circonferenza; onde il detto taglio determinarà la grandezza d’un lato del
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662 triangolo, la quale replicata due volte su pel cerchio, chiuderà la figura e
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663 determinarà gli angoli. </s>
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664 <s>Ma volendo noi formare il quadrato per trarne gli angoli retti, nella
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665 medesima guisa si prendano vintidue portioni ponendo prima un piè del
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666 compasso di sotto e formando l’arco di sopra, che tagli ‘l cerchio da ogni
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667 parte: e poi ponendolo di sopra per formar l’arco di sotto, che tagli ‘l
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668 cerchio; si truovano due misure che stabiliscano la uguale longhezza di due
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669 lati d’un quadrato, onde segue la grandezza degli altri. </s>
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670 <s>Con 14 portioni si truova un arco che tagliando ‘l cerchio stabilisce un lato
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671 del pentagono onde risultano cinque angoli ottusi. </s>
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672 <s>Con 9 l’ottangolo, con 12 l’esagono, con 10 e _ la figura di sette angoli. </s>
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673 <s>E così si potrebbe con diverso numero di parti del diametro truovare il
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674 tagliamento del cerchio per formar ciascuna maniera di figure moltilatere,
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675 per costituir le diversità degli angoli: o in specie, come retti, ottusi e
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676 acuti: o in grandezza, come più o meno acuti o ottusi; che i retti non
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677 variano nella grandezza già mai, essendo sempre uguali; son ben diversi di
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678 specie dagli altri. </s>
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679 <s>Ma proponiamo l’essempio sensato, acciò più facilmente s’intenda la pratica
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680 di questo tagliamento del cerchio. <lb></lb>//<lb></lb>AB. è ‘l diametro diviso in
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681 24 parti uguali. </s>
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682 <s>CD è un arco che costituisce un lato del triangolo. </s>
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683 <s>EF. GH. sono due archi che formano un quadro. </s>
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684 <s>IK. è un arco che determina due lati del pentagono. </s>
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685 <s>LM. un altro che determina due lati dell’esagono. </s>
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686 <s>NO un altro che stabilisce due lati dell’ettangolo. </s>
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687 <s>PQ un altro che forma due lati dell’ottangolo. </s>
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688 <s>Di modo che congiognendosi in qualsivoglia tagliamento del cerchio i punti
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689 de’ tagli si formano le figure di molti lati, e si stabiliscano anchora gli
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690 angoli. </s>
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691 <s>Con la division del diametro si divide ‘l cerchio dallo ‘ntero diametro
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692 perciòchè sempre si taglia per mezzo ad angoli misti nella superficie piana,
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693 e ad angoli retti sferici nella superficie sferica, sì come è ‘l tagliamento
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694 <pb pagenum="folios 11v-12r"></pb>che si fa dall’equinottiale, che divide
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695 tutta la Sfera in due parti uguali, come dice Giovanni Sacrobosco nel cap.
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696 2° della Sfera. </s>
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697 <s>Ma in apparenza sono misti; perciochè quantunque l’equinottiale sia un
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698 cerchio, ansi uno de’ cerchij maggiori della Sfera, contuttociò per la
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699 lontananza, essendo parallelo al diametro e per la dirittura con esso
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700 insieme con l’occhio nostro, per ragion di Prospettiva, si mostra il
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701 diametro stesso della sfera. </s>
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702 <s>Perciochè, talvolta, secondo Prospettiva lo spatio di qual si voglia cosa si
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703 vede in una linea, cioè per la digradatione e scemamento all’apparenza
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704 ridotto ad una linea. </s>
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705 <s>Come suol fare il Sole collocato nel cerchio equinottiale, il cui centro
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706 della circonferenza dell’equatore, e ‘l cui raggio dallo stesso piano
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707 giognendo al centro del Mondo non si ripiega per far conio, ma si ferma
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708 secondo ‘l suo movimento in esso, cioè descrivendo il cerchio stesso
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709 equatore, poiché fra la medesima circonferenza e ‘l centro del Sole è per
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710 diritto del centro del Mondo. </s>
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711 <s>Ma ciò si può vedere nella terza prop. della Gnomonica del Padre Cristoforo
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712 Clavio: overo ciò accade perché ‘l cerchio equatore rappresenta una
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713 superficie, che taglia ‘l corpo sferico passando pel centro; che i corpi non
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714 si dividono se non dalla superficie immediatamente e per cagion di essa
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715 dalla linea anchora: e le linee da punti, e le superficie dalle linee; onde
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716 ‘l taglio della detta superficie sarà fatto dal diametro. </s>
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717 <s>Tal che in apparenza gli angoli che dal suo tagliamento son prodotti, si
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718 mostrano parte rettilinei e pate curvilinei; che nella circonferenza della
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719 Sfera col mezzo del taglio fatto dalla superficie divisa dal suo diametro,
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720 si costituiscano dal diametro stesso, che è una linea retta, e della
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721 circonferen <lb></lb>//<lb></lb>za del detto cerchio posto a dirittura del diametro,
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722 che in apparenza è una linea retta. </s>
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723 <s>Oltre che i paralleli considerati da’ Cosmografi, sono cerchi fra loro
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724 ugualmente lontani, che dividono la circonferenza della sfera ad angoli
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725 retti, e con tutto ciò si descrivano in forma di linee rette per imitar
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726 l’apparenza loro. </s>
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727 <s>Onde, non fuor di proposito, sono appellati questi cerchi paralleli retti,
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728 come si vede appo ‘l Clavio, nella Gnomonica, ed appresso Teodosio nella
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729 prop. quindicesima del primo lib. benchè per altra ragione da esso sieno
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730 appellati retti. </s>
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731 <s>Si divide oltre acciò dalla metà del diametro, quando già formato cerchio, si
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732 applica sei volte sopra la circonferenza, e si taglia il cerchio in sei
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733 parti uguali, che la metà del diametro è la sesta parte della circonferenza:
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734 e con questo tagliamento si forma l’esagono, figura di sei angoli . Onde per
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735 questa divisione si producano gli angoli ottusi nella portion minore del
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736 cerchio. </s>
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737 <s>E queste son tutte le differenze de’ tagliamenti del cerchio, per le quali si
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738 formano varie specie di figure rettilinee, che non sono senza le diverse
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739 maniere d’angoli; e però mentre il vario tagliamento del cerchio c’insegna a
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740 formar diverse ragioni di figure, ci mostra anchora il nascimento degli
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741 angoli diversi; onde con ragione possiamo affermare, che per la varia
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742 division del cerchio si truovano tutte le specie degli angoli. </s>
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743 <s>Dalle cose dette si prende occasion di dubbitare che se ‘l triangolo è
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744 principio d’ogni composition di figura rettilinea, sì come dice il
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745 Commandino, nella Prop. trentaduesima del primo di Euclide, perciochè
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746 ciascuna si risolve ne’ triangoli, come ‘l composto naturale secondo ‘
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747 Filosofi si risolve nelle parti componenti, che sono gli elementi della cosa
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748 composta; <pb pagenum="folios 12v-13r"></pb>per qual cagione s’è dimostrato la
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749 division del cerchio esserne principio, mentre costituisce diverse specie
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750 d’angoli? Si risponde che n’è principio ‘l triangolo e ne è principio ‘l
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751 tagliamento del cerchio, o per dir meglio il cerchio variamente diviso; ma
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752 in diversa maniera. </s>
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753 <s>Perciochè ‘l cerchio tagliato è principio della figura, in quanto stabiliti
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754 diversi tagliamenti nella sua circonferenza ugualmente lontani fra loro; e
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755 di esso molte portioni uguali scambievolmente per la congiuntion de’ detti
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756 tagli fatta con più linee rette, da esso risulta qualunque figura e per
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757 conseguenza ogni maniera d’angolo: e ‘l triangolo è principio dell’altre
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758 figure, cioè non della formatione, ma della compositione e congiognimento
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759 delle parti della figura già formata; e che ciò sia vero vi si scorgano
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760 virtualmente i triangoli, come si vede nel quadrato e nell’altre figure. </s>
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761 <s>Né per questo dobbiamo dire il triangolo esser figura simplice perfetta e
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762 indipendente; da cui piglino origine tutte l’altre; perciochè ‘l triangolo
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763 non dà ‘l nascimento delle figure; ma l’accozzamento delle parti componenti
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764 nelle quali esse si risolvano.</s>
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765 </p>
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766 <p type="main">
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767 <s></s>
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768 </p>
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769 </chap>
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770 <chap>
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771 <p type="head">
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772 <s>D’altri tagliamenti del cerchio onde risultano diverse ragioni d’angoli</s>
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773 </p>
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774 <p type="head">
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775 <s>Cap. 5</s>
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776 </p>
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777 <p type="main">
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778 <s>I tagliamenti del cerchio nel precedente cap. dimostrati si fanno dalla linea
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779 retta e dalla scambievole intersegation de’ cerchij; onde scaturiscono
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780 diverse maniere d’angoli. </s>
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781 <s>Ed oltre a questi altri ne sono, che si eseguiscano dalle superficie coniche,
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782 cagionati dal percotimento de’ raggi del <lb></lb>//<lb></lb>Sole sopra la terra e
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783 da essi parimente nel taglio del cerchio si costituiscano varie specie
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784 d’angoli. </s>
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785 <s>Di questi ragionaremo hora in questo luogo per maggior notitia della
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786 propositione del precedente cap. </s>
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787 <s>Le superficie coniche formate da’ raggi del Sole, essendo lui collocato fuor
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788 dell’equatore, che hanno per punto verticale commune il centro del Mondo,
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789 tagliano diversamente ‘l cerchio Meridiano e in uno stesso tempo
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790 costituiscono gli angoli perciochè o noi consideriamo i cerchij che sono le
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791 basi delle dette superficie esser in fra loro paralleli e retti e tagliare
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792 ancho ad angoli retti sferali: overo consideriamo gli angoli formati dal
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793 contatto de’ lati del conio insieme con la base, e così ritruovaremo gli
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794 angoli acuti: o veramente que che son formati dal concorso e dal contatto
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795 de’ raggi del Sole nel centro del Mondo, come in loro orizzonte; ed in
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796 questo modo trovaremo gli angoli talvolta acuti e talvolta ottusi, secondo
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797 che ‘l centro del Sole sarà più vicino o più lontano all’equinottiale e al
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798 centro del Mondo, come si può osservare nella Gnomonica del Clavio in uno
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799 Analemma descritto per dimostrar la terza Prop. </s>
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800 <s>Così anchora dal segamento delle superficie coniche, e dal piano
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801 dell’Horiuolo parallelo al maggior cerchio, si può trarre ‘l nascimento
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802 degli angoli, sì come si può vedere appo ‘l medesimo Autore nella quinta
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803 Prop. </s>
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804 <s>Perciochè se noi riguardiamo bene, vedremo, che nel tagliamento di due piani
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805 paralleli, che sono due cerchi, fatto da un gran cerchio, si costituiscano
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806 gli angoli <pb pagenum="folios 13v-14r"></pb>retti; quantunque si mostrino acuti
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807 overo ottusi; ma si dee ciò intendere, supposto che l’occhio nostro sia nel
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808 centro del Sole, che co’ suo’ raggi forma le superficie coniche. </s>
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809 <s>Il qual cerchio girando intorno all’asse della Sfera, forma un triangolo,
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810 tagliando la linea retta del piano dell’altro cerchio maggiore, che è uguale
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811 ad esso, la qual linea è asse di lui; e così ancho la settion conica è ‘l
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812 piano dell’Horiuolo, e ‘l piano ch’è base del conio inferiore con una
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813 settion commune; onde si vede sorger la settion detta Parabola, che si forma
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814 ad angoli retti; perciochè si fa col mezzo delle linee rette, e dell’asse
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815 che è perpendicolare. </s>
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816 <s>Di modo che rispetto al triangolo noi truovaremo la costitution degli angoli
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817 acuti. </s>
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818 <s>Similmente nella settione Iperbole e nella Ellipse truovaremo ‘l nascimento
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819 degli angoli retti, acuti e ottusi, come si può vedere appresso ‘l Clavio
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820 nela Prop. sesta e settima del medesimo libro. </s>
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821 <s>In un’altra guisa si ritruova ‘l nascimento degli angoli, cioè col partimento
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822 del cerchio in trecentosessanta gradi; onde nella circonferenza con la
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823 misura d’un arco si può truovare la misura della longhezza d’un lato di
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824 qualsivoglia figura; perciochè ‘l quadrato ha un arco di 90 gradi; il
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825 triangolo di 120; il pentagono di 72 e così del lato di qualunque altra
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826 figura si dee dire; di maniera che truovato un lato, sono anche truovati gli
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827 altri; perciochè un lato solo più volte applicato nella circonferenza è
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828 misura di tutti; poiché si mol <lb></lb>//<lb></lb>tiplica tanto che si chiuda la
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829 figura e dal congiognimento di tutti i lati risultano gli angoli o retti, o
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830 acuti, o ottusi. </s>
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831 <s>Talchè finalmente si conclude, per le cose dette, che da tutti questi diversi
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832 tagliamenti del cerchio hanno origine tutte le maniere degli angoli, il che
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833 è quanto nel precedente cap. si prese a ricercare. </s>
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834 </p>
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835 <p type="main">
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836 <s></s>
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837 </p>
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838 </chap>
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839 <chap>
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840 <p type="head"><s>Se ogn’angolo sia divisibile</s></p>
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841 <p type="head"><s>Cap. 6</s></p>
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842 <p type="main"><s>Se si riguarda al 4° Problema del primo d’Euclide, non havremo
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843 dubbio, che ogn’angolo si divida; poiché quivi così afferma. </s><s>Divider
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844 per mezzo un angolo rettilineo dato. </s><s>E per dimostrare il problema
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845 forma una descrittione con un angolo acuto incontro al quale costituisce un
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846 altro angolo acuto d’un triangolo equilatero collocato sopra la linea retta
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847 del taglio. </s><s>Onde se l’angolo acuto che è minor d’ogn’altro si divide,
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848 è cosa ragionevole che si possa dividere e con più facilità l’angolo ottuso
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849 e ‘l retto; perciochè Euclide dicendo divider per mezzo ogn’angolo
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850 rettilineo, non intende solamente dell’acuto, ma anchora dell’ottuso e del
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851 retto poiché l’angolo rettilineo è genere e l’acuto e l’ottuso e ‘l retto
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852 son le specie. </s><s>Oltre che quel che Euclide mostra nell’angolo acuto
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853 col mezzo del triangolo equilatero o con l’equicrure. </s><s>L’amendue hanno
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854 gli angoli acuti, si può mostrar col mezzo del triangolo ottusangolo e del
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855 rettangolo, come se due squadre s’intersegassero insieme, ma con lo scaleno
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856 non sai. </s><s>Ma se riguardiamo a quel che ne dice Teone o pure Euclide
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857 stesso nella sua Prospettiva nella notation e sopra ‘l 3° Teorema vedremo
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858 l’angolo esser indivisibile. </s><s>Poiché dice l’angolo del contatto esser
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859 indivisibile, citando la decima del 3°. Onde conclude la grandezza <pb pagenum="folios 14v-15r"></pb>che si vede sotto così fatto angolo non
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860 potersi vedere sotto angolo minore. </s><s>Questo, a parer mio si è detto o
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861 da Euclide o da Teone, non per mostrar che l’angolo del contatto
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862 geometricamente non si possa dividere, ma per dimostrar che se la grandezza
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863 fosse collocata sotto un angolo minore di quello del contatto, non si
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864 potrebbe vedere. </s><s>Overo diciamo questa esser un’altra questione
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865 appartenente agli angoli curvilinei, uno de’ quali è l’angolo del contatto.
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866 </s><s>Ma noi qui favelliamo degli angoli rettilinei. </s><s>E rimirando a
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867 quel che sopra ciò dica ‘l Commandino, intenderemo che per la istitutione
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868 elementare non si può divider per mezzo ogn’angolo, che questo da lui si
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869 pone in dubbio e si certifica poi nella specie degli angoli acuti, dicendo
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870 potersi divider in tre parti l’angolo retto, ma non già l’acuto. </s><s>Di
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871 modo che l’istitutione elementare che forza che ogn’angolo non si divida e
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872 per conseguenza l’angolo acuto esser indivisibile, non è altro che le specie
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873 degli angoli sieno elementi e parti componenti altri angoli di specie
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874 diversa da loro; tal che quegli angoli che d’altra specie d’angoli son
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875 composti nelle medesime specie dividendosi si risolvano delle quali composti
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876 sono; ma gli angoli acuti non son composti d’altra maniera d’angoli diversa
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877 da loro adunque non si potran risolvere in altra specie d’angoli adunque
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878 neancho si possan dividere; perché oltre agli angoli acuti non ci dà altra
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879 specie. </s><s>E questa si è la ragione onde ‘l Commandino si mosse a
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880 dibbitare se ogn’angolo si dividesse e ad affermare che gli angoli acuti non
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881 si possan dividere. </s><s>Ma tal ragione si potreb<lb></lb>//<lb></lb>be per
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882 avventura rispondere in questa guisa. </s><s>Altra cosa è la divisione,
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883 altra la risolution delle cose, quando si dice che l’angolo ottuso si divide
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884 in un retto in un acuto overo in più angoli acuti; si dee intender che tal
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885 partimento sia piutosto risolution nelle parti componenti, che divisione.
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886 </s><s>Quando si dice l’angolo acuto dividersi, all’hora s’intende il
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887 partimento esser veramente divisione e non risolutione; perciochè è più
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888 universale la divisione, che la risolutione; perché la risolutione è
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889 solamente quella che distruggendo discioglie le parti componenti, che prima
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890 erano insieme collegate e continue; onde poi risulta la cognition delle
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891 parti e degli elementi, che concorreno a formare il tutto: e questi sono
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892 specie diverse dalla specie della cosa, che di essi è composta. </s><s>La
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893 divisione poi non solamente comprende la risolutione, ma tutte le maniere
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894 de’ separamenti delle cose congionte, o appartenghino alla sustanza del
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895 composto o alla quantità, o alla qualità, o alle diverse nature, o a generi
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896 ed alle specie. </s><s>Però ‘l partimento degli angoli acuti è solamente
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897 divisione non risolutione in altre specie d’angoli; poiché, essendo divisi,
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898 non si risolvano in altri angoli che sieno elementi loro. </s><s>Perciochè
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899 gli angoli, che sono elementi degli angoli, o sono acuti, o son retti; ma
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900 gli elementi degli angoli acuti non si danno; onde avviene che essi non si
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901 possino risolvere negli elementi componenti; e per questo il tagliamento
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902 degli angoli acuti non sarà risolutione, ma simplice divisione. </s><s>Oltre
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903 acciò sì come un tutto composto è realmente diverso dalle parti componenti,
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904 come tutto l’ottusangolo è diverso dal retto e dall’acuto, o da più acuti,
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905 che ‘l compongono; e così l’angolo <pb pagenum="folios 15v-16r"></pb>retto dagli
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906 acuti che ‘l costituiscono; così le parti e gli elementi, dico i detti
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907 angoli componenti sono da esso diversi realmente e ciò si manifesta per la
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908 risolutione; per la qual cosa essendo gli angoli ne’ quali l’angolo acuto si
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909 divide acuti, segue che ‘l suo partimento non sia risolutione, poiché sono
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910 della medesima specie; adunque sarà simplice divisione, la quale non
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911 appartiene agli angoli in specie, come la risolutione per la quale si
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912 costituiscano diverse maniere d’angoli; ma appartiene agli angoli in quanto
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913 sono di quantità e di grado differente secondo ‘l difetto e ll’escesso,
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914 questo in acutezza e quello il misura. </s><s>Perciò che gli angoli acuti si
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915 dividono in angoli acuti, che son diversi da tutto l’angolo solamente
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916 nell’esser più acuti e nell’haver misura più stretta; che l’angolo intero
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917 acuto è ‘l doppio di due, come si può ritrarre dalla dimostratione XI del
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918 IIII di Euclide, o triplo di tre, o quadruplo di quattro angoli acutissimi.
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919 </s><s>Che quantunque l’angolo sia acuto, con tutto ciò, sempre si può far
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920 più acuto, sì come l’ottuso sempre si può far più ottuso; perciochè queste
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921 ragion d’angoli sono imperfette e indeterminate. </s><s>Ma l’angolo retto
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922 non può già mai farsi più o men retto, che essendo retto e ‘n quanto retto è
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923 invariabile e non ha bisogno di perfettione. </s><s>Oltre acciò la divisione
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924 o più tosto la risolutione degli angoli si mostra simile a quella del genere
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925 nelle specie e delle specie negli individui, dove (secondo Platone) bisogna
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926 fermarsi, come se gli angoli acuti sieno simiglianti agli individui; che sì
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927 come gli individui non si risolvano in altra specie; ma si dividano in parti
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928 componenti integrali, così gli angoli acuti non si risolvano in altra specie
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929 d’angoli; ma in parti integrali, che son della medesima natura.
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930 </s><s>Perciochè <lb></lb>//<lb></lb>gli angoli retti e gli ottusi, che per natura
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931 loro sono interi, hanno simiglianza di generi, e gli angoli che sono
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932 elementi e parti componenti hanno simiglianza di specie; ma gli angoli
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933 acuti, presi separatamente, non essendo interi non possono haver simiglianza
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934 di generi, e perciò non si possan risolvere in altre specie. </s><s>Hanno sì
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935 bene apparenza di specie e perciò si dividono in parti, che rassomigliano
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936 gl’individui. </s><s>Di più il ripartimento degli angoli sta in quella
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937 guisa, che si divide un arbolo; perciochè esso si divide nel tronco e ne’
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938 rami e ‘rami non si dividano in altri rami, ma in particelle di rami, che si
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939 appellano ramuscelli: e ‘l tronco non si divide in altri tronchi, ma in
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940 parti di esso, le quali non son il tronco. </s><s>E gli angoli acuti altresì
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941 non si dividano in altra specie d’angoli; che non sono altri angoli oltre ad
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942 essi; ma si dividano in parti, overo in angoli più acuti ed acutissimi.
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943 </s><s>E questa si è diversa divisione da quella degli altri angoli; poiché
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944 la division degli altri angoli costituisce angoli diversi in specie
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945 dall’angolo diviso, come gli acuti da’ retti, e gli acuti e retti dagli
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946 ottusi; ma dalla division degli acuti non si costituiscano angoli diversi di
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947 specie, ma differenti in quantità e in grado. </s><s>Ma in altro modo si può
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948 pruovare ogn’angolo esser divisibile, e questo si eseguisce in tre maniere:
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949 la prima si prende dalla proprietà di chiuder lo spatio; la seconda
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950 dall’esser parte di quantità continua e di spatio terminato da varie figure
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951 angolari; la terza dall’esser diferente dal punto. </s><s>Tutta la
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952 difficoltà è collocata nell’angolo acuto; che degli altri non si dubbita se
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953 sieno divisibili, e perciò tutto ‘l discorso, come da principio si può
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954 vedere infine a questo luogo, si raggira intorno all’angolo acuto, e così
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955 quindi infine al fine, seguirà la medesima intentione. </s><s>E per
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956 cominciare dirò. Se l’angolo acuto fusse indivisi <pb pagenum="folios 16v-17r"></pb>bile non sarebbe niente differente dal punto
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957 il che non essendo vero, non sarà vero l’angolo acuto esser indivisibile.
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958 </s><s>E vero che sì come ‘l punto è principio della linea, così l’angolo
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959 della figura; perciochè non solamente le dà la forza, ma ancho la
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960 denominatione; ma non è vero che sì come ‘l punto di maniera è principio
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961 della linea, che non divien già mai parte di essa, così l’angolo sia
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962 principio della figura; perciochè è parte di essa, e che ciò sia vero,
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963 levisi alla figura rettilinea l’angolo, tosto vedremo essa non solamente
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964 scemarsi, ma distruggersi. </s><s>Si ‘ngannano coloro che fanno l’angolo
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965 acuto indivisibile, non distinguendolo forse dal ponto e credendo che ‘l
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966 ponto del contatto di due linee sia l’angolo. </s><s>Ma non il ponto è
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967 l’angolo, ma ‘l contatto e ‘l concorso di due linee in un medesimo ponto,
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968 come si è già mostrato nel cap. 3. e non può farsi questo concorso se da una
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969 banda le linee non sono separate l’una dall’altra, altramente non sarebbero
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970 due linee, ma una sola, o una cadente sopra l’altra, terminando ne’ medesimi
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971 punti in quella guisa che Euclide dimostra nella IIII del primo.
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972 </s><s>Adunque ‘l punto è termine del contatto, e l’angolo non è altro che
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973 ‘l contatto delle linee, che si fa nel modo già detto. </s><s>Dunque, per
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974 esser termine, non può far che l’angolo acuto non si divida. </s><s>Da
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975 queste ragioni si ritrahe ‘l punto e l’angolo esser cose differenti, come ‘l
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976 termine è differente dalla cosa terminata: e da questa differenza si
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977 conclude l’angolo esser divisibile. </s><s>Ma passiamo più avanti. </s><s>La
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978 medisima ragione hanno le parti che ha ‘l tutto, sì come è sentenza commune
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979 de’ Filosofi, adunque se l’angolo acuto è parte della figura o dello spatio
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980 figurato, che è divisibile, ancho esso sarà divisibile. </s><s>Non si può
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981 negare che la figura sia divisibile; perciochè è quantità continua, la quale
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982 per Aristotile, nella <lb></lb>//<lb></lb>Filosofia Naturale, è sempre divisibile: e
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983 conferma con la commune autorità de’ Matematici e de’ Filosofi; perciochè
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984 alla quantità continua convengon le tre misure. </s><s>Che l’angolo sia
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985 parte della figura, oltre che la sperienza &#039;l mostra, l’afferma Averroe nel
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986 5 della Metafisica, dicendo: l’angolo è parte della figura col mezzo della
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987 quantità e della qualità e si conferma da Henrigo Glareano nel capitolo
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988 primo della Geografia, il quale dice l’angolo esser una particella della
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989 figura, che dal contatto della linea sorge nella larghezza. </s><s>Essendo
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990 adunque l’angolo acuto parte della figura, per essempio del triangolo, e la
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991 figura terminando lo spatio, segue che l’angolo acuto sia anchora parte
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992 dello spatio, e posta la figura divisibile, e lo spatio divisibile, l’angolo
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993 acuto anchora necessariamente sarà divisibile, il che si pruova; perciochè
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994 Euclide in molte dimostrationi del primo, dove si fa comparatione de’ lati
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995 del triangolo, e degli angoli, si dice sempre. </s><s>Angoli sottoposti a
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996 lati uguali. </s><s>Onde si ritrahe che l’angolo non sarebbe stato
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997 sottoposto a’ lati, se in fra essi non si trovasse lo spatio: né i lati si
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998 possono chiamar lati, se toccandosi insieme in un punto (non per diritto di
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999 ciascuno) non determinano la figura o quadrata o triangolare e perciochè
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1000 diventano lati costituendo la figura con gli angoli di essa. </s><s>Però,
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1001 dovunque è angolo, è anchora spatio, ed ogni spatio è divisibile, adunque
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1002 ciascun angolo è divisibile, adunque anchora l’angolo acuto sarà divisibile.
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1003 </s><s>Perciochè onde avviene che l’angolo si divida, se non perché si
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1004 divide lo spatio? Finalmente onde procede la verità dell’ultimo Assioma del
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1005 primo d’Euclide. </s><s>Che due linee rette non chiudino <pb pagenum="folios 17v-18r"></pb>spatio se non dal non formare angolo alcuno? E
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1006 che ciò sia vero si osservi Euclide che si dichiara espressamente, dicendo,
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1007 due linee rette; che se havesse detto simplicemente, due linee non chiudono
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1008 spatio, non sarebbe vero; che due linee curve chiudono spatio, formando la
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1009 figura binangola; poiché si toccano e formano due angoli. </s><s>Adunque ‘l
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1010 chiuder lo spatio nasce dalla costitution degli angoli adunque dagli angoli
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1011 nasce lo spatio rettilineo e curvilineo anchora. </s><s>Però se avverrà che
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1012 si divida lo spatio si divideranno anchora gli angoli qualunque sieno.
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1013 </s><s>AGGIUNTA (1)</s></p>
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1014 <p type="main"><s>Vediamo ora se da altri luoghi di Euclide si può ritrarre, che
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1015 ogn’angolo sia divisibile. </s><s>Nella trentaquattresima del primo afferma,
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1016 che gli spatij parallelogrammi si segano per mezzo dal diametro, il quale
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1017 taglia ancho gli angoli pel mezzo: e nella descrittione si formano acuti.
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1018 </s><s>Dove dal segamento degli angoli si conclude ‘l segamento dello spatio
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1019 parallelogrammo; così convertendo questo detto si può affermare, che dal
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1020 segamento dello spatio parallelogrammo si può venire al segamento degli
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1021 angoli opposti. </s><s>La qual cosa serva in confermation delle cose dette.
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1022 </s><s>Nella dimostratione undicesima del 4° si conclude due angoli di un
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1023 triangolo equicrure simile a un altro triangolo dato, esser doppij di un
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1024 angolo del medesimo triangolo posto nella cima, cioè ciascuno per se stesso
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1025 esser doppio del detto angolo: e non si mostra altramente, che col segamento
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1026 degli angoli, che sono acuti. </s><s>Così anchora nella dimostratione della
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1027 dodicesima si vede un angolo acuto diviso esser doppio d’un altro acuto,
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1028 tutti collocati nel centro d’un cerchio, intorno al quale si è descritto un
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1029 pentagono. </s><s>Nella prop. terza del 6° suppone, che un’angolo </s></p>
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1030 <lb></lb>//<lb></lb><p type="main"><s>AGGIUNTA (1) [Oltre acciò habbiamo ‘l Padre Clavio
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1031 dalla nostra, il quale nello scholio sopra la IX del I dice: quindi è
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1032 manifesto che l’angolo rettilineo si può dividere in quattro angoli uguali,
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1033 in 8, in 32, in 64 e così negli altri sempre procedendo per l’agumento del
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1034 doppio. </s><s>Ma Euclide none insegnò in verun luogo in che modo si potesse
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1035 dividere un angolo rettilineo in tante parti uguali, in quante altri voglia;
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1036 perciò che fine al suo tempo non era stato dimostrato. </s><s>E pertanto,
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1037 per dimostrarlo ci serviremo della dimostratione di Pappo Alessandrino, il
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1038 quale prende per mezzo una certa linea curva, nel fine del sesto libro.
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1039 </s><s>Ma se fra tanto con l’operatione del compasso vorremo dividere
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1040 qualunque angolo rettilineo si propunga l’angolo BAC. da dividersi in cinque
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1041 angoli uguali. </s><s>Dal ponto A. centro si descriva l’arco BC. il quale
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1042 tagli le rette linee AB. AC. in qualunque intervallo. </s><s>Si divida
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1043 l’arco BC. con le seste in tante parti uguali, in quante si ha da dividere
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1044 l’angolo BAC e per essempio si divida in cinque parti uguali ne’ ponti D. E.
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1045 F. G. Hora se dal centro A. a’ detti punti si tirino le linee rette, sarà
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1046 diviso l’angolo BAC. in cinque parti uguali, essendo uguali gl’intervalli
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1047 presi col compasso, cioè BD. DE. EF. FG. GC. saranno tutte queste linee
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1048 uguali. </s><s>Adunque i due lati BA. AD. del triangolo BAD. l’uno all’altro
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1049 saranno uguali, essendo tutti tirati dal centro per la quindicesima def. del
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1050 primo d’Euclide. </s><s>E la base BD. e la DE. essendo uguali, come s’è
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1051 detto; gli angoli DAB. ed EAD. saranno uguali e per la medesima ragione
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1052 l’angolo EAD. è uguale all’angolo EAF. e così gli altri. </s><s>Adunque
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1053 tutti gli angoli appresso all’A. sono fra loro uguali per la ventisettesima
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1054 del 3°; perciochè le circonferenze BD e DE etc. si son poste uguali.
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1055 //]</s></p>
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1056 <pb pagenum="folios 18v-19r"></pb><lb></lb>//<lb></lb><p type="main"><s>d’un triangolo si seghi
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1057 pel mezzo. </s><s>Dunque lo suppon per noto, o come già dimostrato; in
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1058 qualunque modo diciamo, sempre potremo quindi ritrarre ogn’angolo esser
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1059 divisibile, non esser cosa dubbiosa. </s><s>E ‘l Commandino stesso dice
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1060 l’angolo dividersi perché è grandezza non lineale; onde molto meno sarà
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1061 simile al punto. </s><s>Oltre acciò dice anchora, l’angolo esser divisibile,
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1062 atto a ricever l’ugualità e la disugualità secondo la quantità che in esso
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1063 si truova; el parlar suo è indifferente, e perciò non esclude gli angoli
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1064 acuti. </s><s>Nell’ultimo luogo bisogna vedere, se l’oppinion del Commandino
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1065 si può difender in qualche modo. </s><s>E però si osservi, che altro è dire
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1066 assolutamente l’angolo acuto esser indivisibile; altro è dire esser
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1067 indivisibile, cioè che appena si possa dividere per la sua molta strettezza,
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1068 in mezzo alla quale, benchè con grande esquisitezza si tirino le linee; con
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1069 tutto ciò, con grandissima fadiga si possan produrre in maniera le linee in
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1070 mezzo, che non si confondano, e non si congiongano con l’angolo: e ciò
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1071 avviene nelle linee disegnate e negli angoli sensati e materiali. </s><s>Ciò
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1072 s’intende sempre degli acuti, che i retti e gli ottusi, benchè materiali e
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1073 sensati, si possan divider per mezzo come si ritrahe da Euclide nella
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1074 Prospettiva degli specchi, nella dimostrazione del 20° Teorema. </s></p>
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1075 <p type="main"><s>Ma negli angoli non sensati né materiali, ma intelligibili e
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1076 puramente matematici, sempre sarà vero, che ogni angolo sia divisibile: e
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1077 truovandosi angolo quanto più acuto e quanto più stretto si possa
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1078 immaginare, sempre si truovarà ancho una linea che ‘l tagliarà per mezzo.
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1079 </s><s>Ed è questo negotio come quello della Sfera che tocchi ‘l piano in un
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1080 punto; là dove la Sfera materiale nel piano materiale nol fa; ma amendue
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1081 puri matematici sempre mostran vera questa prop. </s><s>Ma ritornando al
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1082 nostro proposito, diciamo intendendosi nel primo modo non esser vera
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1083 l’oppinion del Commandino. </s><s>Ma intendendosi nel 2° esser vera; ma non
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1084 è conforme al puro Matematico, né allo stile <pb pagenum="folios 19v-20r"></pb>d’Euclide; perciochè tutto quel che considera ne’ libri degli Elementi il
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1085 considera, come cosa pura matematica, non vi adattando conditione alcuna,
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1086 altramente non si porrebbe differenza fra le scienze pure matematiche, e fra
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1087 quelle che son miste, o scienze di mezzo.</s></p>
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1088 <p type="main"><s></s></p>
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1089 </chap>
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1090 <chap>
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1091 <p type="head">
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1092 <s>Se si dà l’angolo indeterminato sì come il determinato</s>
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1093 </p>
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1094 <p type="head">
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1095 <s>Cap. 7</s>
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1096 </p>
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1097 <p type="main">
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1098 <s>L’angolo determinato non è altro che ‘l retto; perciochè aggiognendosi o
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1099 scemandosi non è più retto; che se si aggiogne diventa ottuso, e se si
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1100 diminuisce, diviene acuto. </s>
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1101 <s>Però ragionevolmente si dice determinato; che non richiede gionta né
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1102 scemamento: e sì come quelle cose che variano nella specie per la mutatione,
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1103 si dicano determinate, così l’angolo retto si appella determinato; perché si
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1104 muta in altra specie, essendo aggionto o scemato. </s>
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1105 <s>Oltre acciò si dice determinato; perciochè nel definir Euclide gli altri
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1106 angoli, dice l’angolo acuto esser minore e l’ottuso maggior del retto; ma
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1107 dell’angolo retto non si dice esser maggiore o minore; ma uguale; perciochè
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1108 avanti all’angolo retto non si dà altra maniera d’angolo alla quale si possa
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1109 assimigliare. </s>
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1110 <s>Di modo che gli altri angoli si determinano nel retto, ma ‘l retto non si
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1111 determina in altri angoli. </s>
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1112 <s>Gli angoli indeterminati sono gli acuti e gli ottusi; perciochè, secondo
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1113 Proclo, l’angolo retto si forma dalla ragione che nasce dal fine e l’ottuso
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1114 e l’acuto da quella che procede all’infinito. </s>
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1115 <s>Questi si appellano indeterminati perché sempre si possano diminuire od
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1116 accrescere; talchè per questo sono indeterminati; perché non è determinata
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1117 la grandezza loro, sì come per questo si dice l’angolo retto determinato,
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1118 perché è determinata la sua grandezza. </s>
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1119 <s>Oltre acciò si dicano indeterminati; perciochè, secondo Proclo, hanno gran
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1120 forza di accrescere e diminuire gli spati; onde per essi sono indeterminati. </s>
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1121 <s>Ma si avvertisca che quando si dice l’angolo ottuso indeterminato; perciochè
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1122 sempre si può aggiognere <lb></lb>//<lb></lb>e l’acuto altresì; perché sempre si può
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1123 scemare, non si dee intendere che l’uno infinitamente si aggionga e l’altro
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1124 infinitamente si scemi; perciochè o lo scemamento o ‘l crescimento non può
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1125 proceder in infinito; poiché se si cresce in infinito l’ottuso
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1126 necessariamente bisogna venire a un caso, che non vi sia più angolo;
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1127 perciochè quanto più si cresce l’angolo, tanto più si avvicina a perdersi. </s>
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1128 <s>E però per forza di Prospettiva, e per ragion di lontananza dalla vista
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1129 l’angolo ottuso si perde, e si converte in una linea: e quanto più si scema
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1130 l’angolo, tanto più si accosta al non esser più angolo; perciochè finalmente
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1131 si perde sì come al suo luogo ne ragionaremo a pieno. </s>
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1132 <s>Onde quando diciamo l’ottuso sempre potersi crescere, e l’acuto sempre
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1133 scemare, bisogna intender secondo qualche tempo determinato; di modo che
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1134 sempre non voglia dir altro che molte volte si possa o l’uno aggiognere o
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1135 l’altro scemare. </s>
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1136 <s>E di questo per hora non diremo altro.</s>
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1137 </p>
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1138 <p type="main">
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1139 <s></s>
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1140 </p>
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1141 <p type="main">
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1142 <s>Se l’angolo si riduca al tutto alla pianezza e ugualità della linea retta,
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1143 </s>
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1144 </p>
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1145 </chap>
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1146 <chap>
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1147 <p type="head">
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1148 <s>overo alla curvità della circolare</s>
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1149 </p>
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1150 <p type="head">
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1151 <s>Cap.8</s>
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1152 </p>
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1153 <p type="main">
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1154 <s>Per ritruovar la verità di questo problema bisogna considerare, che l’angolo
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1155 può esser riguardato in due maniere, in quanto si truova in un piano, overo
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1156 in una linea piana: e in quanto si truova nella superficie d’una Sfera, o
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1157 nella linea circolare, sì come si vede nelle figure moltilatere. </s>
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1158 <s>Onde considerato l’angolo come in superficie sferica, o ‘n linea circolare,
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1159 noi ‘l vedremo che si ridurrà alla linea circolare o all’incurvamento della
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1160 Sfera: e considerato l’angolo come in un piano, o ‘n’una linea piana il
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1161 vedremo ridursi allo stesso piano, overo alla stessa linea piana, la qual
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1162 cosa si può confermar con Marsilio Ficino nel cap. 41° del Timeo di Platone,
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1163 là dove dice l’angolo ottusissimo esser tosto seguito da <pb pagenum="folios 20v-21r"></pb>due angoli retti; perciochè se oltre a questo
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1164 allongando procederai, subbito perverrai a un piano nel quale cadendo una
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1165 linea retta perpendicolarmente formarà due angoli retti, come si può
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1166 osservare in questo essempio segnato A. Il medesimo per avventura havrebbe
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1167 possuto dire adattando l’angolo ottusissimo alla Sfera, od alla
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1168 circonferenza; perciochè così fatto angolo può esser seguito da due angoli
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1169 retti sferali; che allargandosi continuamente l’angolo ottusissimo posto o
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1170 fuore o dentro la circonferenza, o sopra la superficie della Sfera,
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1171 finalmente bisogna che si riduca alla stessa circonferenza, overo alla
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1172 superficie sferica, come si può osservar nella formation delle figure di
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1173 molti lati, là dove moltiplicati i tagliamenti e costituiti tuttavia archi
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1174 minori, per la trentunesima del 3° d’Euclide si formano sempre angoli
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1175 maggiori ed ognhora più ottusi, come si accenna con questa figura, segnata
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1176 B. Ma per mostrar questo con qualche ragione, bisogna dire che ‘l
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1177 riducimento dell’angolo alla linea si può fare in due maniere, cioè o
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1178 aggiognendo all’angolo o diminuendo. </s>
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1179 <s>Diminuendo si fa tirando sempre nuove linee che si tocchino nel medesimo
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1180 punto dentro alle prime, che chiudono l’angolo proposto, come dentro le AB.
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1181 BC. che formano l’angolo retto ABC. Si tirino le linee DG. che scemandolo il
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1182 fanno diventar acuto; di poi le EF., le quali scemano l’angolo acuto e ‘l
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1183 fanno più acuto di prima; quindi tirate le linee GH, l’angolo scemato
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1184 diverrà ancho più acuto: tirate poi le IK si scemarà l’angolo acuto e si
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1185 farà molto più acuto, e così continuando di tirar nuove linee dentro al
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1186 detto angolo, si tagliarà talmente e talmente si diminuirà che posta in
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1187 mezzo ad esso la linea LB. l’angolo continuamen scemandosi si risolverà in
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1188 essa. </s>
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1189 <s>Perciochè le linee dello scemamento tanto si moltiplicano che si accostano
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1190 alla linea LB e quanto più si accostano tanto più si perde dello spatio; di
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1191 modo che finalmente tutto lo spatio contenuto dall’angolo si converte in una
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1192 linea, quasi nella <lb></lb>//<lb></lb>stessa guisa che fa un piano opposto
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1193 dirittamente all’occhio, come si dimostra dalla Prospettiva. </s>
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1194 <s>Overo si dimostra in questa maniera. </s>
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1195 <s>Essendo già tirato le due linee AB. BC. che terminino l’angolo ABC. si tirino
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1196 le linee DG dentro l’angolo proposto, le quali scemino l’angolo retto e ‘l
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1197 risolvino in acuto: di poi tirate dentro a queste due altre linee, ciò sono
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1198 le EF, si scemi l’angolo acuto DBG. onde divenga via più acuto che ‘l primo
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1199 che sarà l’angolo EBF. e tirate dentro a quest’angolo le linee GH. venga
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1200 scemato l’angolo EBF. e fattone un angolo anchora più acuto che è GBH.
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1201 Tirinsi in fra esso due altre linee IK, necessariamente verrà scemato
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1202 l’angolo GBH e fattone l’angolo IBK. In mezzo al quale si faccia una linea
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1203 retta LB. che divida l’angolo IBK per mezzo; pel 4° problema del primo
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1204 d’Euclide e per le ragioni dette nel cap. 6°. Di modo che si truovi l’angolo
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1205 LBI assai minore. </s>
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1206 <s>Sia l’angolo LBI uguale all’angolo GBC. dividasi dalla linea NB per mezzo;
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1207 adunque ‘l rimanente angolo NBC sarà assai minore d’ogn’altro: taglisi per
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1208 mezzo l’angolo NBC. dalla linea OB. seguirà che l’angolo OBC sia molte volte
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1209 più minore degli altri angoli, essendo minore tante volte dell’angolo
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1210 precedente NBC. il quale è minore d’ogni altro angolo contenuto nella
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1211 division dell’angolo ABC. E se sia possibile con la pratica dividersi
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1212 l’angolo OBC. da una linea retta PB. dico che l’angolo PBC sarà molte volte
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1213 più minore del precedente e de’ rimanenti angoli; onde procedendo
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1214 continuamente in questa guisa, se non con la pratica, almeno con la
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1215 speculativa si divida tante volte l’angolo che si pervenga finalmente alla
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1216 stessa linea, e così si vedrà l’angolo necessariamente in essa essersi
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1217 risoluto; che per la division continua, l’angolo continuamente si ristregne. </s>
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1218 <s>Questo medesimo possiamo dimostrare con la ventunesima del primo d’Euclide là
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1219 dove si mostra che costituite due linee rette dentro un triangolo dato, le
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1220 quali sieno tirate dai termini de’ lati d’esso, essendo minori de’ lati del
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1221 <pb pagenum="folios 21v-22r"></pb>triangolo formano un angolo maggiore; di
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1222 modo che, quante più linee si faranno dentro al triangolo proposto, gli
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1223 angoli da esse contenuti tanto più maggiori diverranno; onde saranno ogni
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1224 volta più ottusi; di maniera che se dentro al detto triangolo continuamente
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1225 si produrranno angoli più ottusi, bisognarà alfine condursi allo svanimento
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1226 dell’angolo e nel riducimento alla linea. </s>
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1227 <s>Perciochè ‘l triangolo è a figura e però è grandezza determinata; onde non si
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1228 può infinitamente scemare: e così gli angoli dentro a essa non possono esser
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1229 infiniti, né infinitamente allargati e fatti ottusi; ma tanto si scema
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1230 l’angolo proposto e tanti angoli si formano dentro ‘l dato triangolo e tanto
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1231 si allargano e si ottusano; che finalmente si perviene al riducimento di
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1232 tutto l’angolo alla linea. </s>
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1233 <s>Ciò non si dimostra essendo provato in gran parte da Euclide nella detta
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1234 prop. </s>
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1235 <s>Solamente per facilitar la intelligenza formaremo qui appresso la figura. </s>
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1236 <s>Ma l’angolo si riduce alla linea retta aggiognendosi in questa maniera, cioè
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1237 costituite due linee che formino l’angolo retto ABC. si tirano di fuore due
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1238 linee rette DE. che terminino nel segno B. e così le IG. HI. e le KL. tanto
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1239 che si pervenga a due linee che sieno per diritto l’una all’altra, ciò sono
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1240 la MB. BN. Sia l’angolo MBA. uguale al NBC. perché dalle cose uguali
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1241 levandosi cose uguali, le rimanenti sono uguali pel 3° Assioma del primo
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1242 d’Euclide. </s>
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1243 <s>Traggasi dall’angolo MBA l’angolo IBC. e dall’NBC. l’angolo GBA. Adunque ‘l
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1244 rimanente HBK è uguale al rimanente IBL. Dividasi l’angolo GBA. e l’HBK.
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1245 dalla linea OP. adunque OBK è uguale all’angolo LBP. Ma l’angolo OBK è
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1246 minore di tutti gli angoli costituiti per le gionte, adunque ancho l’angolo
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1247 LBP. sarà minore altresì di tutti gli altri; ma quello è minore che più si
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1248 ristregne e si accosta alla linea per la ragion precedente; adunque i detti
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1249 <lb></lb>//<lb></lb>angoli son minori e ristregnendosi continuamente si convertano
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1250 in una linea. </s>
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1251 <s>Adunque l’angolo OBK. od altro minore, che per la division si succeda sarà
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1252 fatto la linea MB. e l’angolo LBP. od altro minore, che li succeda col mezzo
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1253 della divisione si sarà convertito nella linea BN. la quale essendo per
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1254 diritto della MB. non conterrà più angolo; adunque l’angolo ABC allargato
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1255 continuamente, overo aggionto con le linee DE. FG. HI. KL. e così seguendo
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1256 finchè la larghezza dell’angolo proposto si accosti alla linea MN. ed
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1257 aggiognendo sempre all’angolo precedente si converte finalmente nella intera
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1258 linea MN. Overo si dimostri così. Aggiontato l’angolo ABC. dalle dette linee
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1259 e diviso l’angolo KBN. dalla linea OB. e l’angolo IBL. della linea BP. segue
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1260 che l’angolo MBK. (essendo già tirata la linea MBN. sia uguale all’angolo
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1261 NBO. Ma l’angolo NBO. è ‘l più piccolo, e ‘l più vicino d’ong’altro a
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1262 convertirsi nella linea, adunque l’angolo MBK. vi sarà ancho vicino essendo
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1263 amendue parti dell’angolo MBH. Così anchora l’angolo NBL. è uguale
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1264 all’angolo LBP. ma l’angolo LBP. è ‘l più stretto e ‘l più vicino
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1265 d’ogn’altro a farsi linea, adunque, adunque l’angolo NBL. vi sarà ancho
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1266 vicino, esendo amendue parti dell’angolo NBI. Ma i detti angoli son parti
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1267 dell’angolo intero KBL. che è la gionta dell’angolo ABC. adunque tutto
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1268 l’angolo intero si è già avvicinato a ridursi alla linea retta MN. alla
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1269 quale all’hora giognerà quando si farà nuova gionta, tanto che si facciano
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1270 due linee che sieno per diritto in fra loro come la MB alla BN. e questo è
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1271 quel che si cercava di mostrare. </s>
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1272 <s>Oltre acciò nella figura che si fa appo Euclide per dimostrar la
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1273 trentacinquesima del primo, la quale è così formata, ponendo due
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1274 parallelogrammi sopra la medesima base, e fra le medesime parallele;
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1275 apparisce un corpo solido con sei angoli e cinque superficie, cioè tre
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1276 parallelogra <pb pagenum="folios 22v-23r"></pb>mme e due triangolari, una delle
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1277 quali superficie parallelogramme si vede apertamente risoluta in una linea e
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1278 due parallele opposte con due altre parallele pure opposte dalle quali è
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1279 contenuto e terminato lo spatio parallelogrammo ADEF. che è una delle dette
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1280 superficie: e così anchora gli angoli del detto spatio si vedano risoluti in
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1281 una linea. </s>
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1282 <s>Ma si può dimostrare in questo modo. </s>
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1283 <s>Sia il lato AB. uguale al lato DC. perciochè ABCD è parallelogrammo, come si
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1284 dimostra da Euclide, el lato AE. è uguale al lato AB. Adunque è ancho uguale
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1285 al lato DC. Ma DC. è lato di parallelogrammo, adunque ancho AE sarà lato di
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1286 parallelogrammo. </s>
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1287 <s>Onde si è già truovato un lato del parallelogrammo. </s>
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1288 <s>Oltre acciò sia il lato FC. uguale al lato BE. per la medesima ragione el
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1289 lato FD è uguale al lato FC. adunque FD è uguale a BE. Ma il lato BE. è lato
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1290 di parallelogrammo, adunque FD anchora sarà lato di parallelogrammo. </s>
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1291 <s>E però si sarà truovato un altro lato del parallelogrammo. </s>
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1292 <s>Onde nella linea AF già sono i lati AE e DF del parallelogrammo. </s>
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1293 <s>I quali son fra loro uguali; perciochè sono uguali a’ medesimi lati de’
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1294 parallelogrammi posti fra le medesime parallele. </s>
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1295 <s>Oltre acciò sia il lato AD uguale al BC. el lato EI. anchora al BC. non
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1296 possono non essere uguali fra loro pel primo assioma d’Euclide. </s>
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1297 <s>Ma il lato BC. è lato di parallelogrammo, adunque AD, ed EF sono due lati di
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1298 parallelogrammo fra loro uguali, come s’è già dimostrato; di maniera che si
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1299 son già truovati due altri lati del parallelogrammo, ciò sono AD. EF.
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1300 Congiongansi i lati AE. DF. FE. DA. e formino quattro angoli ADF. DFE. DAE.
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1301 EFD. si sarà formato il parallelogrammo AEFD uguale a’ parallelogrammi
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1302 rimanenti pel medesimo assioma. </s>
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1303 <s>Ma tutti i lati di questo <lb></lb>//<lb></lb>parallelogrammo sono nella linea AF e
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1304 così gli angoli e per conseguenza ancho lo spatio; adunque il
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1305 parallelogrammo AEDF è tutto nella linea AF. E però sarà vero che la
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1306 superficie parallelogramma e le quattro parallele opposte e gli angoli
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1307 parimente opposti terminanti lo spatio, sieno tutti risoluti in una linea. </s>
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1308 <s>Il che intendevamo dimostrare. </s>
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1309 <s>Nella Prospettiva anchora non è dubbio alcuno, che l’angolo si risolve in una
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1310 linea; perciochè lo spatio di qualunque figura, secondo la position
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1311 dell’occhio nostro talvolta apparisce in una linea, cioè quando l’occhio e
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1312 parallelo all’obbietto piano o quando amendue sono per diritta linea
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1313 opposti; perciochè nella Prospettiva si danno tre positioni dell’occhio,
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1314 cioè o sopra l’obbietto o sotto, o rincontra ad esso e per diritto. </s>
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1315 <s>Quando è sopra l’obbietto, si vede tutto ‘l piano; quando è sotto non si
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1316 vede, ma quando è rincontra per linea diritta, si vede in una linea. </s>
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1317 <s>Ma ritorniamo alle ragion geometriche e diciamo che Euclide nel dimostrar la
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1318 tredicesima del primo mostra la differenza, che è fra le linee che sono per
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1319 diritto, e quelle che non sono, cioè che quelle che non sono, cadendovi
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1320 sopra una linea formano due angoli non uguali a due retti, e per conseguenza
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1321 non uguali fra loro: ed esse formano un angolo ottuso; ma quelle che son per
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1322 diritto insieme, cadendovi sopra una perpendicolare costituiscano due angoli
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1323 uguali a due retti; e uguali fra loro. </s>
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1324 <s>Facciasi il detto angolo più largo, continuamente le due linee verranno per
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1325 diritto e però le due linee diverranno una linea, e l’angolo si risolvarà in
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1326 essa. </s>
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1327 <s>Da queste cose adunque si rende chiaro che l’angolo posto nel piano o nella
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1328 linea retta si risolva nella linea retta e nel piano. </s>
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1329 <s>Hora bisogna vedere se l’angolo posto <pb pagenum="folios 23v-24r"></pb>nella
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1330 superficie sferica e nella circonferenza, si risolva in essa. </s>
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1331 <s>Con la continua division del cerchio e con la diversa forma di figure
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1332 moltilatere, che per l’altra divisione si formano, si può venire in
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1333 cognitione di questo effetto. </s>
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1334 <s>Perciochè in quante più parti si divide ‘l cerchio e di quanti più lati e di
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1335 quanti più angoli si forma la figura, tanto più si accosta alla ugualità
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1336 circolare, sì come di sopra già si è accennato; perciochè multiplicati i
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1337 lati, sempre divengono più corti e quanto più corti, tanto più gli angoli
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1338 sono ottusi; di maniera che col estesimento del numero de’ lati e degli
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1339 angoli, scema la longhezza de’ lati e cresce la larghezza degli angoli; onde
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1340 si fanno tuttavia più ottusi; e finalmente si perdano e si risolvano nella
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1341 circonferenza del cerchio. </s>
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1342 <s>E questo si potrebbe pruovare adducendo l’essempio di tutte le figure che
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1343 dentro o fuor d’un medesimo cerchio si possan formare; cominciando dalla
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1344 figura di tre lati infine a quella di cento; onde vedremo in ciascuna figura
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1345 l’angolo pigliare agumento, ed ognhora più allargandosi accostarsi alla
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1346 tondezza del cerchio o del globo. </s>
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1347 <s>Da tutte queste cose si possan ritrarre due propositioni convesse in fra
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1348 loro, ciò sono:</s>
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1349 </p>
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1350 <p type="main">
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1351 <s>Quanto più si ristregne l’angolo retto, l’ottuso e l’acuto, tanto più si
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1352 accosta alla linea e finalmente in essa si converte.</s>
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1353 </p>
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1354 <p type="main">
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1355 <s>Quanto più si allarga l’angolo retto, l’ottuso e l’acuto, tanto più si
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1356 accosta alla linea, e finalmente divien la stessa linea.</s>
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1357 </p>
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1358 <p type="main">
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1359 <s>Queste medesime propositioni non solamente hanno luogo negli angoli posti nel
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1360 piano; ma anchora in quelli che si pongano nella circonferenza o nella
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1361 sfera. </s>
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1362 <s>Ma ciò basti haver detto intorno a questo proposito. </s>
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1363 </p>
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1364 </chap>
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1365 <p type="main">
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1366 <s>//</s>
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1367 </p>
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1368 <chap>
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1369 <p type="head">
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1370 <s>Per qual cagione alcune volte il cerchio sia detto tutto angolo</s>
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1371 </p>
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1372 <p type="head">
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1373 <s>Cap. 9</s>
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1374 </p>
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1375 <p type="main">
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1376 <s>Se questo detto che’l cerchio sia tutto angolo, fusse assolutamente vero,
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1377 bisognarebbe ancho affermare che fra la curvità della linea circolare e la
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1378 rettitudine della retta fusse qualche proportione; onde avverrebbe che non
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1379 fusse tanto difficile la quadratura del cerchio, se non impossibile. </s>
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1380 <s>Perciochè l’angolo è costituito dal contatto delle linee rette, adunque la
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1381 circonferenza, che costituisce il cerchio doverebbe ad esse esser
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1382 proportionata; che una misura si dice esser tutta un’altra misura, che non
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1383 solo le è proportionata, ma ancho uguale; perciochè per l’ottavo Assioma del
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1384 primo d’Euclide quelle cose sono uguali, che si adattano bene insieme. </s>
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1385 <s>E non si dice una misura esser tutta un’altra se non le si adatta; adunque
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1386 adattandosi l’angolo al cerchio e così per opposito il cerchio all’angolo,
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1387 el cerchio sarà tutto angolo e l’angolo tutto cerchio; che non è altro che
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1388 dire, che sieno uguali: la qual cosa è impossibile. </s>
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1389 <s>Oltre acciò per la terza def. del 5° d’Euclide essendo la proportione una
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1390 certa convenienza di due grandezze del medesimo genere; in quanto appartiene
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1391 alla quantità è cosa certissima, che ‘l cerchio non ha proportione alcuna
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1392 con l’angolo; perciochè e l’angolo e ‘l cerchio sono due generi diversi;
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1393 onde avviene che non abbiano fra loro convenienza alcuna, e perciò non hanno
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1394 proportione. </s>
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1395 <s>E per questo si conclude che non possino adattarsi insieme in maniera che’l
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1396 cerchio vaglia quanto l’angolo e l’angolo quanto ‘l cerchio. </s>
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1397 <s>Quando si dice il cerchio esser tutto angolo, o s’intende esser in quanto
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1398 alla proprietà, o in quanto alla misura. </s>
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1399 <s>Non in quanto alla misura come già s’è pruovato e si pruova di nuovo;
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1400 perciochè una misura si dice tutta un’altra, quando i termini dell’una
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1401 cadano sopra <pb pagenum="folios 24v-25r"></pb>‘termini dell’altra; che tanto è
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1402 longa o larga o alta una cosa quanto è lo stendimento della sua misura, cioè
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1403 di quella linea che si applica alla cosa che si misura secondo la positiva
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1404 delle tre misure della quantità continua. </s>
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1405 <s>Ma ‘l cerchio per questa ragione non si può dir tutto angolo; perciochè se è
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1406 collocato ‘l cerchio dentro all’angolo, l’angolo avanza; se fuore il cerchio
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1407 avanza l’angolo. </s>
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1408 <s>Non in quanto alla proprietà perciochè altre sono le proprietà del cerchio
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1409 altre dell’angolo; che l’angolo comunque si consideri può sempre toccare ‘l
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1410 piano in un ponto e ‘l cerchio non può toccarlo se non considerato in una
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1411 maniera pura geometrica, come s’è detto altre volte: il cerchio è più atto a
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1412 muoversi e forma ‘l movimento continuo ed uniforme; ma l’angolo è
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1413 impedimento al muoversi e se cosa alcuna angolare si muove, l’angolo gli
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1414 interrompe il movimento o ‘l ritarda, o li toglie l’esser continuo uguale e
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1415 uniforme, e fa che ‘l mobile si muova saltando. </s>
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1416 <s>Come dunque sarà vera la propositione che ‘l cerchio sia tutt’angolo? Di più
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1417 se ‘l cerchio fusse tutt’angolo, sarebbe escesso e mancamento di se
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1418 medesimo; ma non è alcuno di questi; perciochè ‘l cerchio è una figura
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1419 perfetta, e finita: e non si dà una figura che avanzi se medesima e da se
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1420 stessa sia manchevole o che sia maggiore o minore in uno stesso tempo e
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1421 secondo uno stesso rispetto. </s>
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1422 <s>Adunque ‘l cerchio non dee chiamarsi tutto angolo, che l’angolo a riguardo
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1423 del cerchio sia escesso di esso o mancamento, si vede per esperienza. </s>
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1424 <s>Però facendosi ‘l cerchio tutto angolo verrà tutto escesso e tutto mancamento
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1425 il quale si dice dell’angolo rispetto al cerchio; adunque verrà fatto tutto
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1426 escesso e tutto mancamento di se medesimo, il che è impossibile. </s>
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1427 <s>Per la qual cosa senza ragione il cerchio sarà appellato tutt’angolo. </s>
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1428 <s>Ma vediamo hora <lb></lb>//<lb></lb>se si può truovar modo da salvar questa
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1429 propositione.</s>
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1430 </p>
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1431 <p type="main">
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1432 <s>Il cerchio potersi dir tutt’angolo si può intender in più maniere, cioè o che
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1433 sia tutt’angolo, cioè che habbia ‘l medesimo estendimento e misura che ha
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1434 l’angolo, e questo è impossibile; poiché ‘l cerchio e l’angolo non son
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1435 proportionali né uguali: o che sia tutt’angolo, cioè in quanto a qualche
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1436 proprietà e simiglianza, e questo è probabile. </s>
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1437 <s>Perciochè se l’angolo raccoglie ristregne e termina lo spatio secondo
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1438 qualunque specie di figura a differenza delle linee rette e parallele che
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1439 non ‘l chiudono, ma ‘l possono allongare in infinito; il cerchio raccoglie,
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1440 ristregne e determina lo spatio seconda la sua specie, mentre è contenuto da
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1441 una sola linea e col mezzo di essa a differenza delle rette linee che nol
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1442 posson terminare. </s>
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1443 <s>Se l’angolo essendo mosso in giro ha proprietà di formare ‘l cerchio; il
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1444 cerchio pel vario e moltiplicato taglio fatto su per la circonferenza,
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1445 produce varie specie d’angoli. </s>
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1446 <s>Se l’angolo prende principio da un punto e in un punto finisce; perciochè ha
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1447 origine dal punto del contatto di due linee, il quale è termine commune di
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1448 esse, o pure congiognimento, e finisce nel medesimo punto; perciò che è
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1449 termine del contatto e per conseguenza dell’angolo; il cerchio comincia da
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1450 un punto, che è quello onde sorge la sua circonferenza, il quale è
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1451 l’istesso, che ‘l termine dell’intervallo: e in un punto finisce; perciochè,
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1452 finito il rivolgimento, quivi termina la circonferenza. </s>
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1453 <s>Overo diciamo che comincia dal centro e nel centro finisce; perciochè ‘l
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1454 nascimento del cerchio procede dal movimento dell’intervallo, il quale
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1455 fissato nel centro, comincia il giramento da esso (che se non havesse
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1456 principio dal centro non potrebbe formare ‘l cerchio) muovendo tutta la sua
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1457 longhezza e con essa anchora il ponto <pb pagenum="folios 25v-26r"></pb>che la
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1458 termina, e ritornando di nuovo nel ponto onde si partì, si ferma appresso
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1459 tal centro, e nel medesimo termine. </s>
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1460 <s>Se l’angolo si fa per contatto el cerchio anchora si fa per contatto;
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1461 perciochè l’angolo si forma per contatto di due linee rette (come più volte
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1462 s’è detto) el cerchio per contatto di linee curve, cioè delle portioni della
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1463 sua circonferenza. </s>
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1464 <s>Overo diciamo che ‘l cerchio sia tutt’angolo, non in atto; ma in potenza,
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1465 cioè che’l cerchio in qualche modo si possa ridurre all’angolo; e ciò dico
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1466 perciochè assolutamente non si può affermare; che non è riducimento
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1467 perfetto, ma vicino al perfetto; che è impossibile a truovarsi; ma ha
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1468 qualche apparenza, sì come dimostraremo al suo luogo: o che sia tutto
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1469 contatto, quasi che sia formato per un continuo contatto delle parti
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1470 succedenti. </s>
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1471 <s>E questo si vede chiaro perché in qualunque parte, col mezzo del tagliamento
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1472 si può ritruovare il punto del contatto. </s>
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1473 <s>Ma avanti che si dimostri il riducimento del cerchio all’angolo, bisogna
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1474 preporre alcune notitie che renderanno più facile la dimostratione di questo
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1475 problema.</s>
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1476 </p>
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1477 <p type="main">
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1478 <s>Ridurre all’angolo il cerchio; onde si possa chiamar tutt’angolo.</s>
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1479 </p>
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1480 <p type="main">
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1481 <s>Suppongansi dunque prima queste positioni.</s>
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1482 </p>
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1483 <p type="main">
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1484 <s>1. In ogni circonferenza è qualche parte di linea, cha ha similitudine con la
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1485 retta, e quanto più la circonferenza è grande, tanto più saranno grandi le
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1486 dette parti.</s>
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1487 </p>
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1488 <p type="main">
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1489 <s>2. Congionte insieme quelle parti della circonferenza che hanno simiglianza
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1490 con la linea retta, necessariamente formano una linea quasi retta.</s>
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1491 </p>
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1492 <p type="main">
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1493 <s>3. Tutti i cerchi fra loro son simili, e così le circonferenze.</s>
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1494 </p>
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1495 <p type="main">
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1496 <s>4. I cerchi minori hanno corrispondenza a’ maggiori e così le parti.</s>
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1497 </p>
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1498 <p type="main">
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1499 <s>5. Che tutte le parti di diverse circonferenze hanno il lor centro
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1500 particolare.</s>
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1501 </p>
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1502 <p type="main">
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1503 <s>6. Che tutte sono simiglianti in fra loro, e congionte insieme tanto che
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1504 formino l’angolo, tanto spatio abbracciano quan <lb></lb>//<lb></lb>to le linee
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1505 rette. </s>
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1506 </p>
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1507 <p type="main">
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1508 <s>La prima positione si potrebbe dichiarar con la minuta division del cerchio e
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1509 con la sperienza del globo della terra e dell’acqua, che apparisce piana, e
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1510 con tutto ciò (come dice il Sacrobosco nella Sfera) vedendonvi nel mare da
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1511 lontano venir una nave a poco a poco, la vediamo quasi sorgere, e ciò accade
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1512 per la tondezza di tutto ‘l corpo dell’acqua e della terra; ma apparisce
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1513 piana; perciochè nelle parti vi è qualche dirittura e pianezza e quella si
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1514 può in parte piana; poiché vi si posano e caminano in piano gli animali e
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1515 perché fra due ponti Zenit e Nadir, detti verticali, cade la linea
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1516 perpendicolare che forma angoli retti nel taglio della palla del mondo
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1517 inferiore: oltre acciò vi si fermano in piano le piante, e gli edificij. </s>
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1518 <s>Ciò si può confermar con Tolomeo nel cap. 10° del 2° dell’Almagesto, là dove
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1519 facendo conferenza delle proportioni degli archi maggiori e minori alle
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1520 corde loro, mostra un arco minore che è una piccola portione d’un gran
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1521 cerchio esser gradi 60 e la sua corda esser parti 0 gradi 60. Si è detto
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1522 nella circonferenza esser qualche parte di linea simigliante alla retta;
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1523 perciochè quantunque sia uguale ed habbia qualche dirittura, con tutto ciò,
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1524 non si può dir linea retta; perché non si produce da un punto all’altro,
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1525 come insegna Euclide nel primo Postulato; ma si produce dal movimento d’un
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1526 punto intorno ad un altro punto immobile col mezzo dell’intervallo, sì come
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1527 si produce tutta la circonferenza.</s>
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1528 </p>
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1529 <p type="main">
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1530 <s>La seconda dichiara, che sì come di più linee rette minori si può formare una
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1531 retta maggiore (che ‘l risolvere è contrario del comporre; onde vediamo, che
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1532 d’una linea maggiore se ne taglia altra minore, com’è manifesto pel 3° Prob.
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1533 del primo d’Euclide), così di più particelle di circonferenza si può formar
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1534 una linea intera quasi retta per la medesima ragione addotta. <pb pagenum="folios 26v-27r"></pb></s>
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1535 </p>
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1536 <p type="main">
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1537 <s>La terza e la quarta si possano dichiarar con la definitione II del 3°
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1538 d’Euclide e con la dodicesima aggionta dal Commandino.</s>
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1539 </p>
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1540 <p type="main">
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1541 <s>La quinta è manifesta; perciochè ciascuna volta che più portioni di
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1542 circonfernza sono accozzate insieme non possono haver un centro commune non
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1543 essendo collocate in guisa che sieno concentriche; ma essendo eccentriche. </s>
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1544 <s>Però necessariamente avviene che ciascuna habbia ‘l suo centro
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1545 particolare.</s>
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1546 </p>
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1547 <p type="main">
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1548 <s>La sesta si può dichiarare per la simiglianza delle parti e degli angoli o
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1549 linee contenute; ansi per la uguaglianza degli angoli che si truovano nella
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1550 circonferenza, come si vede nella XI def. del 3° d’Euclide e nella XII
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1551 aggionta dal Commandino. </s>
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1552 </p>
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1553 <p type="main">
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1554 <s>Proponiamo due essempij d’angoli e ciascuno de’ quali sia adattato al
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1555 cerchio, cioè l’angolo ABC. applicato al cerchio DFE. e l’angolo GFH.
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1556 adattato al cerchio GHI. e formiamo la dimostratione in questo modo. </s>
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1557 </p>
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1558 <p type="main">
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1559 <s>Sieno nella prima figura due linee rette AB. e BC. che facciano l’angolo ABC.
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1560 e dal punto B. del contatto dell’angolo, si tiri una perpendicolare, cioè la
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1561 BF. costituiscasi in essa il punto M.(pel 3° Post. di Euclide) e fatto
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1562 l’intervallo MF. si formi ‘l cerchio DFE che tocchi i termini delle linee
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1563 rette AB. e BC., così sopra il centro M. posto il centro T alto 3 portioni
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1564 della linea MF. si faccia il cerchio GHI. e sopra il punto T pongasi ‘l
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1565 centro R. di uguale altezza, e si descriva il cerchio KML. ed ascendendo
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1566 quasi per due portioni si ponga il centro V e si disegni ‘l cerchio NO e nel
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1567 taglio fatto dal cerchio KLM. nella perpendicolare, dico nel segno S. si
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1568 determini un altro centro e si descriva il cerchio piccolo PQX. di maniera
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1569 che si saranno formati cinque cerchi i quali col convesso della
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1570 circonferenza loro si congiogneranno con le linee <lb></lb>//<lb></lb>AB. e BC.
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1571 formando con esse l’angolo ABC. I quali cerchi sono fra loro proportionali
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1572 perciochè tutti sono di proportione sesquialtera; poiché per la diminutione
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1573 loro si palesa la commune proportione in fra essi. </s>
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1574 <s>Sono ancho simili come è manifesto per la terza positione e come si può
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1575 confermare con la undicesima def. del 3° d’Euclide, perciochè le portioni di
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1576 loro prendono angoli uguali, o sopra esse si fanno angoli uguali; che
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1577 l’angolo ABM. è uguale all’angolo ABC. essendo uguali le basi AM. MC. *(Per
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1578 la contrapositione della ventiquattresima e venticinquesima del primo
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1579 d’Euclide) [nota in margine] <pb pagenum="folios 27v-28r"></pb></s>
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1580 </p>
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1581 <p type="main">
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1582 <s>I detti cerchi diminuiti proportionalmente formano quasi un angolo simile
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1583 all’angolo ABC. Onde essendo tutti simili e prortionali, la medesima
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1584 possanza di formare un angolo converrà communemente a tutti insieme uniti ed
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1585 incatenati e perciò potrà dirsi assolutamente il cerchio farsi tutto angolo;
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1586 poiché le parti di ciascun cerchio concorreno a formarlo: che per la quarta
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1587 positione i cerchi maggiori e minori e le parti loro hanno in fra lor
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1588 corrispondenza, con la quale si accordano a far l’angolo. </s>
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1589 <s>Ma pruoviamo l’angolo costituito da questi cerchi; esser uguale al
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1590 rettilineo. </s>
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1591 <s>Congionti i ponti AM. MC. onde seguono due triangoli ABM. MBC. Perché per
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1592 Euclide, nella diciannovesima prop. del primo al maggior angolo è sottoposto
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1593 maggior lato; per opposito all’angolo uguale sarà sottoposto lato uguale. </s>
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1594 <s>Ma ‘l lato AM. del triangolo ABM è uguale al lato MC. del triangolo CBM.
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1595 Adunque l’angolo ABM. sarà uguale all’angolo CBM. Ma il lato IM. è ancho
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1596 lato del triangolo formato con la forza delle portioni de’ cerchij GKNPB e
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1597 ‘l CM. è lato del triangolo ILOQB adunque l’angolo GBM sarà uguale
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1598 all’angolo IBM. Ma l’angolo GBM è uguale al ABM e ll’angolo IBM è uguale
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1599 all’angolo MBC. Adunque tutto l’angolo IBA è uguale a tutto l’angolo ABC. Ma
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1600 l’angolo IBA è formato di portioni di cerchij, nelle quali si son risoluti,
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1601 e l’ABC. è fatto di linee rette; adunque l’angolo di portioni di cerchij
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1602 sarà uguale all’angolo rettilineo; onde seguirà che ‘l cerchio si sia
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1603 ridotto tutto l’angolo. </s>
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1604 <s>Si potrebbe ancho dimostrare facendo comparation degli spatij contenuti da
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1605 amendue gli angoli; ma passia <lb></lb>//<lb></lb>mo ad un’altra dimostratione,
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1606 nella quale apparisca l’angolo formato di parti di cerchij; onde si formi un
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1607 angolo almeno simile se non uguale interamente all’angolo rettilineo. </s>
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1608 <s>Essendo già proposto il cerchio GHST e in esso tirati il diametro QR. e la
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1609 perpendicolare FI che pasi pel centro, sopra la quale sieno tirati tre
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1610 cerchij minori proportionali, cioè KL. MN. OP. essendo già diviso in quattro
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1611 quarte uguali dal diametro detto e dalla perpendicolare, si tirino due linee
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1612 che dividino pel mezzo le dette quarte, ciò sono GT. HS. onde tutto ‘l
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1613 cerchio maggiore venga diviso in otto parti. </s>
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1614 <s>Si divida la metà del diametro QR. in quattro parti uguali e presa la quarta
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1615 si ponga nell’altra metà del diametro appresso alla circonfrenza nel segno
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1616 QX. come nella prima metà di esso è collocata nel segno VR. Di poi preso per
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1617 centro il punto T. e per intervallo TG. si faccia col compasso una portion
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1618 di cerchio cominciando a muover il piede dal termine G. tanto che ‘l cerchio
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1619 maggiore si congionga col primo minore terminando nel segno W. Così ancho
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1620 posto ‘l centro S. e l’intervallo SH. si formi una portion di cerchio che
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1621 terminando nel segno S congionga il primo cerchio minore col maggiore. </s>
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1622 <s>Quindi preso per centro ‘l taglio V. si faccia con le seste l’arco WF. e
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1623 preso ‘l taglio X. si descriva l’arco SF. i quali archi si tagliaranno nella
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1624 perpendicolare nel punto F e formaranno l’angolo. </s>
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1625 <s>Finalmente stabilito centro il termine della terza portione in ogni metà del
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1626 diametro si formi un arco che agguagli e congionga i due archi GW. WF. ne’
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1627 punti DY. e così un altro che spiani e colleghi i due archi HS.SF. ne’ punti
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1628 LZ. Di modo che di tutte le portioni de’ cerchi e di tutti gli archi si son
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1629 formate due linee quasi rette, cioè GF. FH. le quali toccandosi nel punto F.
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1630 formano l’angolo GFH. uguale all’angolo ABC. la qual cosa si dimostrarà qui
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1631 appresso. <pb pagenum="folios 28v-29r"></pb>Perché le linee GF. FH. sono uguali
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1632 alle linee AB. BC. le quali sono linee rette, segue che ancho le GF. FH.
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1633 sieno rette. </s>
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1634 <s>Perciochè per la prima suppositione son formate di parti di circonferenze di
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1635 cerchij, le quali son rette. </s>
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1636 <s>Oltre acciò per 8° Assioma el primo d’Euclide, quelle cose sono iguali, che
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1637 si adattano bene insieme. </s>
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1638 <s>Ma le linee GF. FH. assai ben si adattano con le linee AB. BC. adunque le GF.
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1639 e FH. sono uguali alle AB. e BC. e le grandezze, che sono uguali fra loro,
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1640 sono ancho simili, essendo della medesima specie; adunque GF. FH. saranno
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1641 simili ad AB. e BC. ma AB. e BC. son linee rette, adunque GF. ed FH. saranno
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1642 simiglianti a linee rette, ma sono ancho linee rette, come s’è dimostrato. </s>
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1643 <s>E con tutto ciò son composte di particelle di circonferenze. </s>
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1644 <s>Hora congionti i punti HA. AF. AG. GF. FH. nella seconda: e nella prima
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1645 congionti CM. MB. MA. AB. BC. si saranno due triangoli, cioè nella prima
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1646 ABM. BCM. e nella seconda FGA. FHA. Perché due lati del primo triangolo
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1647 della prima dico dico BA. ed AM. sono uguali a due lati del 2° BC. e CM.
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1648 segue il triangolo ABM esser uguale al triangolo MBC. Così anchora nella
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1649 seconda figura per due lati del primo traingolo, cioè GF. e GA. sono uguali
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1650 a’ due lati del 2° FH. HA. segue parimente i due triangoli GFA. FAH esser
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1651 uguali. </s>
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1652 <s>Perciochè hanno anchora gli angoli uguali e la base commune. </s>
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1653 <s>Ma mutando ordine il lato GF è uguale al lato AB. come già si è dimostrato el
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1654 lato GA. al lato AM, perciochè *(Per la quindicesima def. del primo
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1655 d’Euclide) [nota in margine] escono del centro di due cerchi uguali; così
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1656 anchora per la medesima ragione il lato FH sarà uguale al lato BC. ed HA. a
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1657 CM. e la base FA: commune alla base BM. Adunque i triangoli FGA. FHA.
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1658 saranno uguali a’ triangoli BAM. BCM. Adunque l’angolo <lb></lb>//<lb></lb>GFA. è
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1659 uguale all’angolo ABM. e l’angolo AFH è uguale all’angolo MBC. Ma gli angoli
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1660 GFA. e AFH son parti dell’angolo GFH. costituito dalle linee formate di
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1661 particelle di circonferenze: e gli angoli ABM. e MBC. sono parti dell’angolo
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1662 ABC. fatto col toccamento di due linee rette. </s>
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1663 <s>Adunque tutto l’angolo GFH. è uguale a tutto l’angolo ABC. ma è ancho simile. </s>
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1664 <s>Adunque l’angolo GFH fatto di linee circolari sarà simile e uguale all’angolo
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1665 ABC. che è retto. </s>
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1666 <s>Il che bisognava dimostrare.</s>
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1667 </p>
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1668 <p type="main">
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1669 <s>E le dette linee circolari, che per la prima supp. son simiglianti alle
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1670 rette, non son separate dal cerchio (perciochè o sono attualmente costituite
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1671 da tagliamenti di quattro cerchij disuguali o sono formate stabiliti altri
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1672 centri, come già si è dimostrato nella pratica) ed essendo congionte insieme
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1673 formano le linee GF. FH. le quali toccandosi nel ponto F. formano l’angolo
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1674 GFH. Adunque insieme col cerchio lo formano. </s>
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1675 <s>Onde segue che ‘l cerchio si riduca all’angolo, mentre le parti della sua
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1676 circonferenza si son fatte linee, che pel contatto formano l’angolo. </s>
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1677 <s>Però segue anchora ‘l cerchio potersi chiamar tutto angolo, perché
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1678 risolvendosi in parti tutto si converte in angolo. </s>
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1679 <s>Si potrebbe ancho dimostrare facendosi comparatione fra i cerchij, cioè fra
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1680 maggiori e maggiori, e fra minori e minori e fra segamenti e fra le portion
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1681 de’ cerchij, e quindi trahendo l’ugualità delle linee che sopra i convessi
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1682 loro son tirate, e di poi l’uguaglianza degli angoli. </s>
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1683 <s>Ma che le linee rassembrino le rette benchè sieno parti di circonferenze,
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1684 quindi è manifesto; perché si formano stabilito ‘l centro, *(come si è
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1685 supposto nella .5. positione) [nota in margine] e perché essendo parti di
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1686 circonferenze di cerchi maggiori hanno qualche rettitudine, come è chiaro
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1687 per la prima positione. </s>
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1688 <s>Che le parti delle circonferenze sieno fra loro simili, come si è supposto
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1689 nella positione terza e sesta <pb pagenum="folios 29v-30r"></pb>ed uguali e
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1690 proportionate quindi si può ritrarre che son linee curve; si perché sopra
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1691 esse si possan formare angoli uguali per la def. aggionta dal Commandino nel
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1692 3° di Euclide. </s>
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1693 <s>Si disse che questo riducimento del cerchio all’angolo non è perfetto,
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1694 riserbandosi la ragione a questo luogo. </s>
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1695 <s>Si ritragga dunque la ragione dalla pratica e dalla teorica di questo
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1696 problema. </s>
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1697 <s>Che ‘l congiognimento de’ cerchij non è tanto esatto che tolga l’angolo
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1698 rimanente in fra’ segamenti che si fanno da essi scambievolmente: nella
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1699 linea, che gli congiogne è talmente diritta e piana, che non habbia qualche
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1700 incurvamento per natura sua, benchè sia incognoscibile. </s>
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1701 <s>O vero diciamo intendersi il cerchio tutto angolo; perciò che l’angolo è più
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1702 vicino a diventar cerchio, che a farsi linea retta; come si vede nelle
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1703 figure di molti lati; che quanto più, che quanto più vi si moltiplicano i
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1704 lati e gli angoli, tanto più l’angolo si spiana, e finalmente diventa tutto
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1705 cerchio. </s>
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1706 <s>E queste son tutte le maniere con le quali per me s’è possuto sostentare che
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1707 ‘l cerchio sia tutto angolo.</s>
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1708 </p>
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1709 <p type="main">
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1710 <s></s>
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1711 </p>
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1712 </chap>
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1713 <chap>
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1714 <p type="head">
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1715 <s>A che serva l’angolo nell’Universo</s>
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1716 </p>
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1717 <p type="head">
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1718 <s>Cap. 10</s>
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1719 </p>
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1720 <p type="main">
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1721 <s>Nell’Universo non si trova cosa che sia indarno come si ritrahe dal Filosofo. </s>
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1722 <s>Però se l’angolo è cosa reale overo intelligibile, bisogna vedere qual fine
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1723 habbia. </s>
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1724 <s>Primieramente per ispedirmi più tosto che sia possibile dirò che sì come ‘l
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1725 cerchio è stato truovato specialmente pel movimento locale circolare (onde
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1726 avviene che ‘l Cielo sia inclinevole al giramento e la sua inclinatione sia
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1727 detta da’ Filosofi principio passivo interno e quasi natura) così la figura
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1728 angolare per la quiete e col mezzo di essa l’angolo anchora; perciochè gli
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1729 angoli impediscono ‘l movimento continuo; che <lb></lb>//<lb></lb>(sì come dice
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1730 Pietro da Medina nel 3° cap. del primo lib. dell’Arte del navigare) la
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1731 propria operatione del Cielo è muoversi continuamente e circolarmente e
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1732 perciò conviene che habbia la figura accomodata a tal movimento e questa si
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1733 è la figura ritonda perciochè è privata d’angoli, che sono impedimento al
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1734 muoversi. </s>
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1735 <s>Oltre acciò non si possano costituir le quattro quarte del Mondo senza gli
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1736 angoli perciochè o prendansi da’ quattro venti, in quanto son cagionati da
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1737 quattro Pianeti secondo gli Astrologi, cioè da Giove, dal Sole, da Marte e
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1738 dalla Luna. </s>
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1739 <s>Onde son detti dalla Luna occidentali da Marte meridionali, dal Sole
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1740 orientali e da Giove settentrionali: overo si prendano da’ quattro
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1741 triplicità del Zodiaco, come dall’Ariete, dal Leone, dal Sagittario, tutti
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1742 di qualità calda e secca, e Segni orientali onde vengono i venti orientali:
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1743 o dal Toro, dalla Vergine, e dal Capricorno, tutti di qualità fredda e secca
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1744 e meridionali; onde nascono i venti australi: o da Gemelli, dalla Libbra e
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1745 dall’Aquario, tutti di qualità calda e humida e occidentali; onde procedono
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1746 i venti occidentali di maniera che queste son quattro triplicità di Segni,
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1747 appellate ignea, terrea, aerea, aquea, le quali costituiscono quattro venti,
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1748 che danno nome a quattro parti e a quattro angoli del Mondo; perciochè come
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1749 dice l’istesso autore nel 5° cap. la tondezza della terra ha quattro parti,
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1750 angoli e regioni principali non solamente avvertite da’ Filosofi e dagli
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1751 Astrologi, ma ancho dalla Sagra Scrittura, come appo San Luca nel 14° cap. e
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1752 appresso David nel Salmo 106. E queste si nominano e conoscano pe’ quattro
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1753 venti cardinali, ciò sono Levante, Ponente, Tramontana, Mezzodì. Da’ quali
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1754 tirate <pb pagenum="folios 30v-31r"></pb>le linee rette che passino pel centro
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1755 del Mondo, necessariamente tagliano la sua circonferenza ad angoli retti e
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1756 costituiti in fra quattro altri venti collaterali e congionti con le linee
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1757 loro formaranno gli angoli della medesima maniera. </s>
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1758 <s>Così nella carta da navigare i 32 venti che stabiliscono altretanti rombi
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1759 dividono la tondezza del Mondo in tante parti e ne’ tagliamenti
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1760 costituiscono gli angoli, i quali si possano ancho ritruovar nella sua
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1761 circonferenza per cagion delle posture destra, sinistra, e alta e bassa, che
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1762 dal Filosofo sono adattate al Cielo in riguardo della postura dell’huomo,
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1763 come si può vedere nel lib. del Cielo e del Mondo. </s>
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1764 <s>Là dove la parte destra del Cielo è detta oriente e la sinistra occidente, e
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1765 l’alta e la bassa i due poli. </s>
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1766 <s>O vero potremo truovar gli angoli per la differenza del movimento, osservando
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1767 ciò che dice San Tommaso nel 3° della Filosofia Naturale nella particella
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1768 54. Nel cielo sono le differenze del movimento; avanti l’emisfero di sopra,
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1769 dietro l’emisfero di sotto; a destra l’oriente; a sinistra l’occidente; di
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1770 sopra mezzo giorno; di sotto settentrione. </s>
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1771 <s>E ‘l reflettimento de’ raggi del Sole sopra la terra, formando diversità
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1772 d’angoli, produce più o men gradi di calore, secondo che con l’angolo si
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1773 unisce e si disgrega la forza del lor calore; onde nasce ‘l vario
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1774 temperamento delle stagioni dell’anno. </s>
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1775 <s>Perciochè ‘l ripiegamento de’ raggi è cagione del calore; perciochè molti
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1776 raggi si riducano in un punto vero, dove per la caldezza unita e rinvigorita
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1777 il luogo si riscalda e s’abbracia, sì come disse Alberto Magno nel tratt. 3°
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1778 del 2° del Cielo. </s>
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1779 <s>Nel qual luogo bisogna osservare che i raggi della luce si dirizzano a un
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1780 vero punto, <lb></lb>//<lb></lb>cioè non sensibile e materiale, ma geometrico; e
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1781 questo da Alberto si aggiogne per mostrare in quanta acutezza talvolta per
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1782 virtù degli angoli formati dal percotimento e piegamento de’ raggi del Sole;
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1783 si raccolga e si ristrenga il calore; che si uniscano in una cosa
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1784 indivisibile; onde non è maraviglia, che la virtù si multiplichi insieme col
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1785 calore; onde il luogo venga o caldo overo acceso; che perciò disse Averroè
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1786 nella parte 72 del 3° del Cielo, che l’acutezza degli angoli è cagion di
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1787 bruciamento. </s>
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1788 <s>Oltre acciò di tre movimenti locali, che si vedano continuamente
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1789 nell’universo: il retto e ‘l reflesso non potran formarsi già mai senza ‘l
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1790 producimento degli angoli. </s>
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1791 <s>Perciochè ‘l movimento delle cose gravi o delle leggiere si fa sempre per
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1792 linea retta; che mentre le cose gravi si muovano formano una linea che
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1793 ferisce ‘l piano della terra, ad angoli retti, come si può pruovare con
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1794 l’autorità di Tolomeo nel 7° cap. del primo dell’Almagesto e se potessero
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1795 passar la terra la detta linea fermandosi nel centro tagliarebbe ad angoli
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1796 retti il diametro del Mondo. </s>
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1797 <s>Onde con ragione disse Ammonio sopra Porfirio, che gli angoli retti non si
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1798 possan tirare altrove, che al centro. </s>
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1799 <s>E mentre le cose lievi si muovano verso ‘l concavo della luna, fanno una
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1800 linea che divide la circonferenza del convesso della sfera del fuoco e
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1801 ferisce la concavità di quella della luna ad angoli retti, ed amendue le
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1802 linee sono perpendicolari solide; perciochè sopra la superficie nella quale
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1803 cadono formano angoli conseguenti uguali e si formano dentro ‘l corpo
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1804 sferico. </s>
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1805 <s>Il movimento reflesso, o ripiegato che si dica, necessariamente si fa col
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1806 producimento degli angoli in quella guisa che farebbe una palla da giocare
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1807 cadente <pb pagenum="folios 31v-32r"></pb>sopra ‘l piano, la quale tosto
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1808 percossolo risaltesse e tornasse a muoversi al contrario che ‘l movimento
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1809 reflesso, secondo Aristotile nella parte 64 dell’8° della Nat. </s>
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1810 <s>Filosofia è composto di contrarij movimenti; i quali si discernano per la
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1811 quiete infra posta e la quiete pel percotimento della cosa mobile nel piano,
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1812 e ‘l percotimento per l’angolo si conosce; onde il Filosofo stesso nella
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1813 settion 16 parte 4 de’ Problemi disse: quelle cose che cadono nella terra
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1814 formano angoli nel piano da ogni banda del punto nel quale habbiano toccato
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1815 ‘l piano perciochè tutte per lor natura si muovano per linea retta; ma
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1816 quelle che desiderano muoversi a un luogo uguale, quando dalla linea
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1817 perpendicolare e del diametro feriranno nel piano, perciò faranno col
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1818 balzamento loro angoli uguali perché devidono in parti uguali ‘l diametro. </s>
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1819 <s>Ma quelle cose che per lato cadono, perché feriscono la terra non per linea
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1820 perpendicolare ma da un punto sopra accade che ribattute dal luogo si
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1821 muovano nella contraria parte. </s>
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1822 <s>Ma per qual cagione Aristotile mostra che le cose che fanno ‘l movimento
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1823 reflesso si muovano dalla perpendicolare e dal diametro? Non solamente per
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1824 la ragione detta, ma per accennare che qualunque cosa, che si muova verso ‘l
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1825 piano, mentre a poco a poco si stacca dalla linea perpendicolare, si parte
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1826 ancho dal diametro; perciochè ovunque sia, sempre costituisce una linea che
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1827 se fusse continuata, e passasse pel centro col mezzo della cosa mobile grave
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1828 e quindi si allongasse fine alla parte opposta necessariamente formarebbe ‘l
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1829 diametro. </s>
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1830 <s>E che ‘l reflettimento non si faccia <lb></lb>//<lb></lb>senza angoli si conferma col
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1831 detto del Commentatore nella parte 49 e nel .C. 61 del 2° del Cielo. </s>
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1832 <s>Che ‘l reflettimento non si fa se non secondo gli angoli terminati. </s>
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1833 <s>Ma poiché si è ragionato de’ movimenti non par fuor di proposito ragionar
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1834 ancho di quelle cose che gli facilitano. </s>
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1835 <s>Gran differenza è da una cosa mobile di figura piana ad una di figura
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1836 angolare, e così da una che habbia angoli retti overo ottusi ad una che gli
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1837 habbia acuti. </s>
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1838 <s>Perciochè le cose che sono di figura piana, quantunque sieno gravi
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1839 difficilmente fendono ‘l mezzo e perciò con difficoltà si muovano, il che si
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1840 vede per la tardanza del movimento loro: e poste sopra l’acqua stanno a
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1841 gallo invece di calar nel fondo. </s>
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1842 <s>Le cose angolari, quantunque minori e talvolta ancho men gravi, più
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1843 facilmente tagliando l’aria, si muovano verso la terra: e nell’acqua tosto
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1844 cadano nel fondo. </s>
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1845 <s>Quelle che hanno angoli retti e ottusi tagliano sì bene l’aria e per essa si
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1846 muovano scendendo al lor centro; ma non senza qualche resistenza del mezzo. </s>
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1847 <s>Finalmente le cose che hanno angoli acuti, facilmente trapassano ‘l mezzo, e
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1848 per esso si muovano velocissimamente giognendo al centro. </s>
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1849 <s>Ma se per avventura prenderemo due cose gravi o sieno angolari, o sieno
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1850 ritonde o qualunque altra figura; ma sieno disuguali di grandezza e di peso
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1851 e da qualche luogo eminente le lassaremo cadere in un medesimo tempo in
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1852 terra, onde avviene che amendue vi arrivino in un medesimo tempo?Dovrebbe
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1853 pur giognervi più presto la minore poiché occupando meno spatio e havendo
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1854 men diametro è più atta a fendere ‘l mezzo; tanto più che truova minor <pb pagenum="folios 32v-33r"></pb>resistenza per la sua poca larghezza: overo vi
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1855 havrebbe prima ad arrivar la maggiore, come più grave; poiché quanta più
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1856 gravezza vi è tanta più velocità di movimento vi doverebbe essere. </s>
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1857 <s>A questo si dee dire che amendue in uno stesso tempo arrivano al piano della
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1858 terra perciochè se nella minore non è tanta gravezza che possa dar tanta
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1859 forza al movimento vi è la brevità della figura in supplimento; talchè le
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1860 toglie la resistenza dell’aria e le dà la facilità al trapassarla: e se
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1861 nella maggiore è tanta larghezza che dà occasione alla troppa resistenza
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1862 dell’aria vi è la gravezza che che supplisce; di modo che la poca grandezza
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1863 e la men resistenza dà alla minore quella parte di virtù che le mancava per
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1864 muoversi: e la gravità della maggiore supplisce la parte della forza, che
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1865 dalla figura e dalla resistenza veniva scemata. </s>
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1866 <s>Di maniera che amendue supplimenti agguagliano le forze d’amendue, onde
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1867 vengano a muoversi e ad arrivare in un medesimo tempo in terra. </s>
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1868 <s>Oltre acciò l’acqua che ha da muover qualunque maniera di macchina non ha
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1869 forza nel muovere se cadendo non fa angoli simiglianti a’ retti; perciochè
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1870 se scorre per linea obliqua vien trattenuta a poco a poco dalla superficie
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1871 del piano che sia ritorto onde perde in gran parte l’impeto del corso; ma se
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1872 cade a piombo segue col’impeto suo naturale fine al fondo della fossa, anzi
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1873 dall’angolo della caduta piglia forza maggiore; poiché quivi diviene
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1874 abbandonata da ogni maniera di sostentamento e lassata tutta nel dominio
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1875 della sua gravità. E per trapassare a essempij più nobili
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1876 <lb></lb>//<lb></lb>l’huomo anchora, o stando in piedi o sedendo forma l’angolo;
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1877 perciochè essendo creato di figura diritta per sua natura costituisce la
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1878 linea perpendicolare, la quale cadendo nel piano del suo posamento forma da
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1879 ogni banda l’angolo retto, non solamente stando in piedi, ma sedendo
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1880 anchora, sì come si può vedere appresso Arist. nella question 30. delle
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1881 Meccaniche, là dove si mostra, che sedendo si costituisce in due luoghi
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1882 l’angolo retto e stando in piedi in un sol luogo e la ragione si è perciochè
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1883 l’angolo retto, che è uguale ovunque sia è cagione della quiete e dello
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1884 stare in piedi, e quest’angolo è prodotto dalla linea perpendicolare non
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1885 sopra ‘l pavimento uguale, ma sopra la circonferenza della terra, e per
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1886 conseguenza da una linea che ha riguardo al centro di essa, ond’avviene che
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1887 ‘l detto angolo sia cagion di quiete; perciochè è formato da una linea che
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1888 si termina nel centro, che è la cagion principale della quiete. </s>
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1889 <s>Nella stessa maniera e per la medesima cagione le piante stanno fisse in
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1890 terra, formando angolo retto, o nel terreno piano per natura o nel piano del
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1891 suo posamento, cioè quando sono in spiagge o su per un colle o vero in un
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1892 terreno al tutto erto e scosceso. </s>
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1893 <s>Perciochè quantunque la terra sia obliqua, con tutto ciò le piante stanno
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1894 sopra essa diritte; di maniera che tutte in quanto all’apparenza feriscono
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1895 ‘l terreno ad angoli impari, ottusi e acuti; ma in quanto appartiene al
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1896 natural posamento loro feriscono la terra ad angoli retti, di modo che
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1897 ciascun arbolo per se stesso col suo diritto nascimento riduce in piano e
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1898 livella il terreno vicino nella stessa guisa che si suol fare misurando le
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1899 superficie delle colline, tenendo in aria la misura da una banda <pb pagenum="folios 33v-34r"></pb>e ponendo in piano continuamente e tirando le
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1900 linee a piombo da uno estremo della misura sopra la terra, cme se havessemo
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1901 a formar molti gradi uguali. </s>
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1902 <s>Di maniera che da tutto questo discorso possiamo trarre l’uso dell’angolo
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1903 esser collocato in molte parti dell’Universo, cioè o rispetto al movimento
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1904 delle parti loro o rispetto alla quiete e al posamento e a’ compartimenti
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1905 della Sfera del Mondo fatti secondo la position de’ venti: o riguardando
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1906 l’effetto de’ raggi solari: o le differenze del luogo e del movimento: o
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1907 pure la differenza delle figure delle cose che si muovono verso ‘l centro e
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1908 finalmente l’impeto e la forza delle cose cadenti.</s>
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1909 </p>
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1910 <p type="main">
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1911 <s></s>
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1912 </p>
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1913 </chap>
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1914 <chap>
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1915 <p type="head">
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1916 <s>Quali sieno gli usi dell’angolo nella Geometria, nell’Astronomia, nella
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1917 Prospettiva</s>
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1918 </p>
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1919 <p type="head">
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1920 <s>Nella Geometria</s>
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1921 </p>
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1922 <p type="head">
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1923 <s>Cap. 11</s>
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1924 </p>
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1925 <p type="main">
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1926 <s>In tutta la Geometria gli angoli son di tanta virtù che senza essi non
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1927 solamente non potrebbe fabbricar le figure e ‘corpi regolari o irregolari;
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1928 ma ancho quando pur ciò potesse fare, non potrebbe formarne dimostratione
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1929 alcuna, per pruovar la nesità delle propositioni alle figure e a’ corpi
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1930 appartenenti. </s>
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1931 <s>Perciochè chi bene osservarà Euclide, vedrà non esser quasi dimostratione
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1932 alcuna, che non prenda forza d’illatione dalla conferenza degli angoli. </s>
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1933 <s>Ma questa fin qui è la minor parte del giovamento, che la Geometria riceve
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1934 dall’uso degli angoli. </s>
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1935 <s>Però per darne più copioso ragguaglio, in quanto comporta questo luogo;
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1936 aggiognerò che gli angoli son <lb></lb>//<lb></lb>cagione di molti effetti;
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1937 perciochè o sono l’origine o ‘l termine delle figure e de’ solidi: o da essi
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1938 depende l’accrescimento e la diminutione degli spatij e delle figure e de’
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1939 corpi solidi: o l’accrescimento o lo scemamento de’ lati e delle basi: o per
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1940 essi si conoscano le linee parallele, o le perpendicolari, o le linee per
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1941 diritto. </s>
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1942 <s>E per esplicar a parte a parte tutti questi effetti degli angoli senza i
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1943 quali la Geometria sarebbe vana, con ordine retrogrado mi farò da
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1944 quest’ultimo, il quale si manifesta da Euclide nella quattordicesima prop.
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1945 del primo, cioè che per gli angoli si venga in cognitione di due linee, che
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1946 sieno poste per diritto fra loro. </s>
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1947 <s>Perciochè conosciuto che due linee rette tirate da diverse parti a una linea
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1948 retta, e ad un ponto dato in essa, cioè ad uno de’ suoi termini, formano due
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1949 angoli conseguenti uguali a due retti, anzi due angoli retti stessi; tosto
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1950 si conosce che le dette linee son per diritto fra loro. </s>
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1951 <s>E che questo sia vero, quindi apparisce, che quando gli angoli che da esse si
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1952 costituiscano non sono uguali a due retti, e amendue retti le dette linee
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1953 non possono esser per diritto fra loro, sì come chiarissimamente apparisce
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1954 nella dimostration d’Euclide nel luogo citato. </s>
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1955 <s>Di modo che sì come delle linee poste per diritto fra loro, le quali
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1956 convengono nella stremità d’una linea retta che stia a piombo nascono gli
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1957 angoli conseguenti non solamente uguali a due retti, ma ancho retti, così
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1958 all’incontro, da questi angoli conseguenti retti e uguali a due retti si
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1959 costituiscano le linee per diritto fra loro. </s>
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1960 <s>E ciò avviene perciochè la linea alla quale e dal ponto della quale da
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1961 diverse bande son tirate altre linee è perpendicolare, ed esse, essendo per
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1962 diritto formano una stessa linea retta piana, che <pb pagenum="folios 34v-35r"></pb>determina i detti angoli. </s>
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1963 <s>La notitia degli angoli è ancho cagione che si conoschino le linee
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1964 perpendicolari; e questo si può vedere manifestamente nella decima def. del
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1965 primo d’Euclide e nella prop. dodicesima. </s>
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1966 <s>Perciochè nella definition degli angoli retti si mostra che dalla costitution
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1967 degli angoli segue la linea, che sta sopra a un’ altra linea retta esser
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1968 perpendicolare; che facendo angoli conseguenti e dalle bande uguali, gli
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1969 forma ancho retti; poiché l’esser uguale è proprio degli angoli retti, ed
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1970 essendo retti necessariamente segue la linea che sovrasta alla giacente
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1971 esser perpendicolare. </s>
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1972 <s>E che questo sia vero rimiriamo con diligenza la definitione.</s>
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1973 </p>
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1974 <p type="main">
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1975 <s>“Quando la linea retta stando sopra un’altra fa gli angoli da lati fra loro
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1976 uguali, sono amendue retti e la linea che sta sopra si chiama perpendicolare
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1977 a quella a cui sovrasta”.</s>
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1978 </p>
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1979 <p type="main">
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1980 <s>Questa definitione, al parer mio, procede in questa maniera; perciochè è
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1981 composta di due parti, una si è ‘l caso che si propone, e l’altra si è tutto
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1982 quello che da esso, in guisa di fratto si cagiona. </s>
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1983 <s>Primieramente adunque propone. </s>
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1984 <s>Quando la linea retta, stando sopra un’altra, fa gli angoli da lati fra loro
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1985 uguali. </s>
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1986 <s>E questo si è ‘l caso che si compone d’una parte indeterminata e universale,
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1987 che è una linea retta star sopra un’altra: ed una parte che è ‘l
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1988 ristregnimento e la condition che determina la prima, cioè fare gli angoli
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1989 da’ lati fra loro uguali. </s>
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1990 <s>Di poi immediatamente aggiogne. </s>
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1991 <s>Sono amendue retti e la linea, che sta sopra si chiama perpendicolare. </s>
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1992 <s>E questa si è la seconda parte, che nasce dalla prima, come ‘l frutto dalla
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1993 pianta, la quale ancho è composta di due <lb></lb>//<lb></lb>portioni, la prima che
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1994 detti angoli sieno amendue retti, la seconda che nasce dalla prima, che la
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1995 linea che le sovrasta sia perpendicolare. </s>
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1996 <s>Onde si vede chiaramente che dalla formation degli angoli per la sovrastante
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1997 linea nascono gli angoli retti; e per gli angoli retti s’acquista la
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1998 certezza, che la linea che sta sopra a un’altra sia perpendicolare: e che
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1999 dagli angoli uguali si concludano gli angoli retti e da essi si concluda la
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2000 linea perpendicolare; ma non si potrebbe concludere se non si procedesse
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2001 dalle cose note alle non conosciute, si come è chiaro appresso ‘Filosofi;
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2002 adunque bisogna dire che dalla cognition degli angoli uguali, e retti
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2003 necessariamente si viene alla cognition della linea perpendicolare; come si
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2004 manifesta dalla detta definitione. </s>
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2005 <s>Ma nella prop. dodicesima si vede manifestamente, nel formar la sua
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2006 descrittione dalla forza degli angoli stabilirsi necessariamente la
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2007 perpendicolare. </s>
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2008 <s>E questo più chiaro si vede nella sua dimostratione; perciochè poste due
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2009 linee rette uguali fra loro, le quali sono ancho per diritto ed una commune
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diff changeset
2010 in fra esse, e due linee uguali a due linee, cioè una di quelle insieme con
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diff changeset
2011 la commune a un’altra presa insieme con la commune: così ancho la base
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diff changeset
2012 uguale alla base (perciochè per la descrittion del problema si formano due
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diff changeset
2013 triangoli) seguono due angoli costituiti dalle due rette linee, e dalla
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diff changeset
2014 commune conseguenti e uguali, e la detta linea commune esser quella, che
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diff changeset
2015 sovrasta alla retta linea data, ed esser perpendicolare, come si conferma
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parents:
diff changeset
2016 per la decima def. già addotta. </s>
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diff changeset
2017 <s>Quindi adunque si conclude che truovata <pb pagenum="folios 35v-36r"></pb>l’ugualità degli angoli conseguenti, tosto veniamo a conoscere la
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diff changeset
2018 perpendicolare, la quale dagli antichi fu detta Gnomone; perciochè lo
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diff changeset
2019 gnomone è una linea retta, che cade ad angoli retti sopra l’orizzonte; onde
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diff changeset
2020 vediamo che dal cader ad angoli retti si forma questa linea così detta; che
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diff changeset
2021 se non cadesse ad angoli retti; cioè se cadendo non formasse angoli retti,
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2022 non si potrebbe appellar Gnomone, né perpendicolare. </s>
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diff changeset
2023 <s>Dagli architetti si chiama Catetto, overo Piombo o linea a piombo, la quale
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diff changeset
2024 sopra ‘l piano forma gli angoli a squadra, che geometricamente si chiamano
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diff changeset
2025 retti; che se detti angoli non fossero a squadra la linea non sarebbe a
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parents:
diff changeset
2026 piombo. </s>
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diff changeset
2027 <s>Di maniera che l’esser a squadra sia cagion, che ella sia a piombo e
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parents:
diff changeset
2028 perpendicolare. </s>
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parents:
diff changeset
2029 <s>Ma passiamo più avanti. </s>
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2030 <s>Dalla cognition degli angoli si cagiona la cognition delle parallele sì come
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2031 chiaramente vedrà chiunque osserva Euclide nel 5° postulato e dalla prop.
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parents:
diff changeset
2032 ventisettesima, infine alle trentaduesima. </s>
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2033 <s>In tutte queste propositioni e nelle dimostrationi loro, dalla ugualità degli
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diff changeset
2034 angoli alterni, degli esterni, e degli interni e opposti e dall’esser uguali
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parents:
diff changeset
2035 a due retti si ritrahe la cognition delle parallele. </s>
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diff changeset
2036 <s>Il che si vede osservando gli angoli che si costituiscano da una linea, che
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parents:
diff changeset
2037 sia tirata sopra le parallele in tal guisa, che le taglino ad angoli retti. </s>
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parents:
diff changeset
2038 <s>La qual cosa potrà da ciascuno esser intesa in Euclide nelle dimostrationi
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parents:
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2039 delle accennate prop. </s>
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2040 <s>Ma nel postulato 5° dagli angoli fatti per la linea cadente sopra due rette
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parents:
diff changeset
2041 linee, i quali son minori di due retti, s’impara a conoscer qua’ linee non
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2042 sieno <lb></lb>//<lb></lb>parallele, di maniera che, per opposito, se la linea
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parents:
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2043 cadente sopra esse formasse gli angoli uguali a due retti, noi dalla notitia
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parents:
diff changeset
2044 di quest’angoli potremo cavar la cognitione delle parallele. </s>
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diff changeset
2045 <s>Gli angoli oltre acciò son cagione dello scemamento, e dell’accrescimento de’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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2046 lati e delle basi delle figure triangolari e parallelogramme così ancho
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2047 della ugualità e della disugualità, come si può vedere appo Euclide nelle
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parents:
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2048 prop. .6., .19., .21. e .24. e .25. del primo. </s>
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parents:
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2049 <s>Perciochè nella .6. da due angoli d’un triangolo si traggano i lati
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parents:
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2050 sottoposti a essi esser uguali, il che è certissimo per la dimostratione; sì
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2051 come pel contrario posti due angoli disuguali d’un medesimo triangolo,
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parents:
diff changeset
2052 anchora i lati diverranno disuguali. </s>
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2053 <s>Nella .19. si mostra, che sotto maggior angolo di ciascun triangolo è
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parents:
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2054 collocato maggior lato. </s>
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parents:
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2055 <s>E ciò non avvien per altro se non perché mentre cresce l’angolo cresce ‘l
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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2056 lato e però nella .18. prop. disse, il maggior lato di ciascun triangolo
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parents:
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2057 esser sotto a maggior angolo. </s>
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parents:
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2058 <s>Però possiamo dire che gli angoli col mezzo della grandezza loro son cagione
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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2059 della grandezza de’ lati. </s>
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parents:
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2060 <s>Si potrebbe ancho ciò ritrarre dalla prop. ventesima. </s>
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parents:
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2061 <s>Ma questo si dee intendere del lato che è opposto all’angolo nelle figure di
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parents:
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2062 tre lati. </s>
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parents:
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2063 <s>Vediamo hora se si può ritrarre ‘l medesimo effetto degli angoli ne’
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parents:
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2064 parallelogrammi. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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2065 <s>Dice Euclide nella prop. trentacinquesima. </s>
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parents:
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2066 <s>I parallelogrammi posti nella medesima base e nelle medesime parallele esser
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parents:
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2067 fra loro uguali: là dove per dimostrarla si forma questa descrittione ABCDEF
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2068 e sopra essa si dimostrano i due parallelogrammi ABCD e BEFC. esser uguali
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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2069 essendo nella medesima <pb pagenum="folios 36v-37r"></pb>base BC e nelle
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parents:
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2070 parallele BC. FA. Ma questo è vero in quanto alla teorica, e secondo la
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parents:
diff changeset
2071 dimostratione fondata in questo supposto, che sieno nella medesima base, e
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parents:
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2072 fra le medesime parallele. </s>
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parents:
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2073 <s>Ma secondo la pratica apparisce ‘l contrario, il che non doverebbe accadere;
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parents:
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2074 perciochè la teorica e la pratica nelle scienze e nell’arti doverebbero
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parents:
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2075 convenire insieme; poiché amendue sono quasi due gambe, con le quali la
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2076 scienza e l’arte procede; che se per avventura mancasse e l’una e l’altra,
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parents:
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2077 andarebbe zoppa. </s>
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parents:
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2078 <s>Ansi benchè vi sieno amendue, con tutto ciò essendo con qualche sproportione,
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parents:
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2079 conviene che amendue in qualche parte vadano zoppicando. </s>
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parents:
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2080 <s>Apparisce dico il contrario perciochè il lato EB. del secondo parallelogrammo
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2081 è maggiore del lato AB. del primo e ‘l lato FC. del secondo è maggiore del
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parents:
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2082 alto DC. del primo. </s>
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parents:
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2083 <s>Perciochè posto ‘l centro B. per la .3. supp. del primo d’Euclide e lo
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parents:
diff changeset
2084 intervallo BE. si descriva ‘l cerchio GE. e posto ‘l centro C. e lo
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parents:
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2085 intervallo CF. si descriva ‘l cerchio FH. vedremo espressamente i detti
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parents:
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2086 cerchi avanzare i lati del primo parallelogrammo e per conseguenza essendo i
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parents:
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2087 due intervalli uguali, che son due lati del secondo parallelogrammo, saran
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parents:
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2088 cagione che ‘l secondo parallelogrammo sia maggiore del primo, che se fosse
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parents:
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2089 uguale il cerchio EG. toccarebbe il lato EC. nel segno E. Di modo che per la
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parents:
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2090 def. del cerchio i lati de’ parallelogrammi verrebbero a esser tutti uguali. </s>
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parents:
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2091 <s>Ma i due lati BE. FC del 2° parallelogrammo son maggiori e contengono angoli
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2092 maggiori; onde dagli angoli maggiori BEF. FCB. dependono i lati maggiori BE.
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2093 FC. che se fussero <lb></lb>//<lb></lb>uguali agli angoli del primo parallelogrammo,
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2094 i lati anchora sarebbero uguali. </s>
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2095 <s>Oltre acciò teoricamente anchora apparisce ciò esser vero, supposti tre
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2096 triangoli uguali, ne’ quali si sieno risoluti i parallelogrammi, cioè ADB.
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2097 DBC. ECF. che per la .19. del primo, sotto a ciascuno angolo maggiore de’
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2098 tre triangoli è posto un lato maggiore; che ciascun triangolo ha due angoli
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2099 minori e un maggiore; ma il parallelogrammo BEFC. è composto di due
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2100 triangoli DBC. ECF., adunque tutto l’angolo BEF. sarà composto dell’angolo
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2101 BEC. minore e dell’angolo CEF.maggiore, e però seguirà che BEF. sia molto
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2102 maggiore dell’angolo BAD., così anchora per la medesima ragione, tutto
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2103 l’angolo BCF. sarà molto maggiore dell’angolo ABC. e così si potrebbe dire
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2104 facendo comparatione di questi angoli maggiori agli altri angoli rimanenti;
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2105 onde seguirebbe che per cagion degli angoli maggiori i lati del secondo
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2106 parallelogrammo sieno maggiori de’ lati del primo. </s>
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2107 <s>Oltre acciò se ‘l lato BE. fusse uguale al lato AB. bisognarebbe affermare
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2108 che ‘l diametro fusse commensurabile col lato del parallelogrammo, la qual
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2109 cosa è impossibile. </s>
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2110 <s>Ma ritorniamo alle figure di tre lati. </s>
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2111 <s>Euclide nella .21. prop. del primo dimostra che dall’angolo maggiore
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2112 contenuto da linee rette tirate dentro un dato triangolo da’ termini di
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2113 esso, cioè dagli angoli presso alla base hanno dependenza a due linee minori
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2114 de’ due lati del triangolo dato; di maniera che si può dire che quanto più
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2115 sarà maggiore l’angolo, tanto più saranno minori i lati, e così per
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2116 opposito; e quanto più saranno mag <pb pagenum="folios 37v-38r"></pb>giori,
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2117 tanto più sarà minore l’angolo e così allo ‘ncontro. </s>
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2118 <s>Perciochè sopra la base del triangolo dato non si può collocar un altro
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2119 triangolo che non sia minore, dico dentro ‘l triangolo dato; ma uguale non
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2120 mai; perciochè sarebbe ‘l medesimo triangolo, come si potrebbe dimostrare
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2121 per la .7. del medesimo libro. </s>
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2122 <s>Di modo che fin qui avemo dimostrato, che dall’esser maggiori gli angoli
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2123 considerati, o come superiori o come accanto a’ lati del triangolo, si fanno
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2124 i lati minori o maggiori. </s>
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2125 <s>Ma vediamo ora se dalla grandezza degli angoli de’ triangoli si possan trar
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2126 le grandezze delle basi. </s>
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2127 <s>Questo è facil cosa a veder; perciochè ‘l dimostra Euclide nella .4. e nella
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2128 .24. e .25. del primo. </s>
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2129 <s>Nella quarta si vede chiaramente; perciochè da’ due lati supposti uguali a’
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2130 due lati, seguendo due angoli uguali a due angoli contenuti da linee rette
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2131 uguali nascono le basi uguali alle basi; ma ciò si certifica per la sua
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2132 dimostratione. </s>
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2133 <s>Nella .24. supposti nella medesima guisa due lati de’ triangoli uguali, ma
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2134 l’angolo esser maggior dell’angolo contenuto dalle linee rette uguali, si
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2135 dimostra le basi de’ triangoli dover esser disuguali, cioè una maggior
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2136 dell’altra, la qual cosa si rende certa per la dimostratione come se
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2137 neccessariamente dagli angoli maggiori e minori contenuti da linee rette
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2138 uguali si produchino le basi maggiori e minori. </s>
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2139 <s>E che questo sia vero si vede espressamente per la venticinquesima che è sua
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2140 conversa. </s>
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2141 <s>Da tutte queste ragioni dunque si mostrano le grandezze maggiori o minori e
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2142 gli agumenti e gli scemamenti de’ lati e delle basi de’ triangoli nascer
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2143 dalle grandezze maggiori o mi <lb></lb>//<lb></lb>nori e dagli accrescimenti e
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2144 scemamenti degli angoli de’ triangoli; così anchora i lati de’
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2145 parallelogrammi farsi maggiori o minori dalla grandezza degli angoli. </s>
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2146 <s>Hora preposte tutte queste cognitioni, sarà facil cosa cercare se sia vero
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2147 che dagli angoli dependa l’accrescimento e la diminution delle figure e
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2148 degli spatij. </s>
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2149 <s>Se è vero che ‘lati, e le basi de’ triangoli all’hora sieno maggiori, quando
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2150 son maggiori gli angoli, all’hora minori, quando son minori gli angoli;
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2151 all’hora uguali quando sono uguali; sarà ancho vero che gli spatij e le
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2152 figure che gli terminano sieno maggiori o minori o uguali secondo che gli
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2153 angoli son maggiori, o minori o uguali. </s>
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2154 <s>Il che espressamente si pruova nella .4a. del primo. </s>
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2155 <s>Oltre acciò, ancho Proclo nel 4° lib. sopra Euclide, nel com. 9° dice che
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2156 l’ugualità e la disugualità degli angoli ha gran forza di accrescere e
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2157 scemare gli spatij. </s>
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2158 <s>E questo avviene perciochè ne’ triangoli quanto più si dà un angolo maggiore,
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2159 tanto più si dà maggior lo spatio: né si può conceder spatio maggiore che
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2160 non si dia ancho almeno un lato maggiore; ma in verità si danno ancho due
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2161 lati maggiori, uno de’ quali sarà sottoposto all’angolo maggiore, e l’altro
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2162 conseguente al primo lato maggiore; anzi anchor esso è sottoposto all’angolo
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2163 maggiore, come è manifesto per la diciottesima e diciannovesima del primo. </s>
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2164 <s>Questo medesimo si potrebbe adattare a’ corpi solidi, perciochè in essi
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2165 anchora si truova l’angolo, che secondo Euclide nella .21. definitione
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2166 dell’11° è differente dall’angolo piano; perciochè è l’inclinatione di più
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2167 di due linee che si toccano non essendo nella medesima superficie: ed è
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2168 compreso da più di due <pb pagenum="folios 38v-39r"></pb>angoli piani, come ‘l
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2169 dimostra nella prop. .21. dell’11°. Di maniera che se si agumentano gli
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2170 angoli piani, le superficie anchora ricevono agumento: e se si agumentano
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2171 amendue, necessariamente si accresce l’angolo solido. </s>
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2172 <s>Ma veniamo hora a considerare se sia vero che gli angoli sieno il termine e
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2173 ‘l principio delle figure piane e solide. </s>
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2174 <s>Avanti che si esplichi se ciò sia vero, fa di mestiero avvertire che da
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2175 questa questione s’escluda la figura circolare, l’ovata, la sferica ed altre
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2176 figure terminate da una linea e da una superficie sola. </s>
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2177 <s>Se è vero che ‘l ponto sia principio e termine della linea, e la linea sia
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2178 principio e termine della superficie, che non è senza figura: e la
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2179 superficie è termine e principio del corpo, come sarà vero che gli angoli
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2180 sieno principio e termine delle figure piane e solide? Bisogna avvertire che
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2181 nella Geometria o ‘l punto, o la linea, o la superfice, o l’angolo possa
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2182 dirsi principio delle figure in due maniere, cioè che sia principio del
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2183 producimento loro considerato assolutamente e ‘n generale, cioè che dal
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2184 movimento loro intelligibile risultino le figure e ‘ solidi: overo che sia
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2185 principio speciale in maniera che per esso si determini la specie della
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2186 figura considerandolo in quanto che senza esso la figura non può haver
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2187 sussistenza. </s>
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diff changeset
2188 <s>Nel primo modo si vede ‘l principio quasi materiale e nel 2° quasi formale. </s>
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2189 <s>Nel primo modo l’angolo non è principio delle figure; ma ‘l punto, la linea e
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2190 la superficie: nel 2° l’angolo solamente è principio di esse. </s>
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2191 <s>Perciochè quel medesimo principio che dà la denominatione alle figure, gli dà
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2192 ancho l’essere; che la de <lb></lb>//<lb></lb>nominatione nasce dalla determination
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2193 della specie e tale determinatione si produce dagli angoli i quali non sono
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2194 altro che la differenza costitutiva e divisiva delle figure. </s>
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2195 <s>Di maniera che l’angolo sia lo stesso atto della figura, il quale non
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2196 altramente che forma ponga separamento le figure. </s>
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2197 <s>Oltre acciò non si possan formar le figure piane, senza chiuder spatio, né le
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2198 solide senza terminar grossezza; perciochè sì come due linee rette non
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2199 chiudono spatio come si vede nella quarta del primo d’Euclide, così due
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2200 superficie non chiudono grossezza, ne si può già mai chiuder spatio alcuno,
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2201 né alcuna grossezza di corpo solido senza formar gli angoli. </s>
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2202 <s>Onde le figure piane e le solide non si dicano figure avanti che sien formati
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2203 gli angoli; ma subbito formati, che tosto ne risultano le specie determinate
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2204 delle figure. </s>
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2205 <s>E però, secondo ‘l numero degli angoli prodotti si specificano e si
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2206 denominano le figure. </s>
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2207 <s>Anzi ancho le linee curve non posson chiudere spatio senza gli angoli, come
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2208 avviene nella figura triangola. </s>
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2209 <s>Ne si possan formar le figure curvilinee se prima non si costituiscano gli
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2210 angoli con l’intersegamento de’ cerchij, come si vede nella prima del 1°.
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2211 Nella medesima guisa si dee dire che l’angolo sia termine delle figure,
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2212 perciochè se produce la figura, la termina anchora; e mentre per l’angolo si
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2213 chiude lo spatio e la larghezza e la grossezza, si termina anchora. </s>
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2214 <s>Overo diciamo l’angolo esser principio delle figure come origine di esse, e
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2215 termine in quanto che per gli angoli vengono specificate. </s>
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2216 <s>Gli angoli oltre acciò stabiliscono l’origine e ‘l termine de’ dia <pb pagenum="folios 39v-40r"></pb>metri de’ cerchi, così ancho delle diagonali,
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2217 che dividono ‘l parallelogrammo: fanno discerner le linee finite
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2218 dall’infinite; perciochè solamente si formano nelle finite le quali hanno i
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2219 punti che le determinano; che non si formano se non pe’l contatto in un
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2220 punto collocato già in esse: e l’angolo non retto ridotto al retto facilita
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2221 la misura delle figure irregolari; come si ritrahe dalla tredicesima del
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2222 primo. </s>
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2223 <s>Gli angoli sono in fra loro misura perciochè l’angolo retto è misura de’ non
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2224 retti, ma non per opposito, se forse non riguardiamo fra i retti e gli
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2225 acuti. </s>
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2226 <s>Percioche ‘l retto è misura dell’acuto; che fattane comparatione si vede
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2227 quanto l’acuto sia minore del retto e così per opposito quanto ‘l retto sia
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2228 maggior dell’acuto: overo si dice l’acuto poter esser misura del retto;
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2229 perciochè nel retto tante volte si replica la sua grandezza, finchè si
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2230 misuri tutto; che ‘l retto è moltiplice dell’acuto e l’acuto come grandezza
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2231 minore lo minore lo misura, come è manifesto per la seconda def. del 5°. Ed
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2232 applicando i detti angoli a’ numeri, facendo l’angolo retto esser quanto ‘l
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2233 numero maggiore e l’acuto quanto ‘l minore, potremo dire l’acuto angolo
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2234 esser misura del retto; perché in quanto numero minore è parte del maggiore
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2235 perciochè lo misura, com’è chiaro per la terza def. del 5° e di questo non
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2236 si dee dubbitare, perché l’angolo retto è composto almeno di quattro angoli
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2237 acuti: e si potrebbe per avventura ciò confermare con la definition del
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2238 misuramento addotta da Niccolò Tartaglia nel primo cap. del suo primo lib.
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2239 di Geometria, cioè:</s>
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2240 </p>
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2241 <p type="main">
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2242 <s>Misurare è un voler truovar quante volte in una quantità si ritruovi alcuna
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2243 quantità famosa, overo alcuna parte: overo sapere quante parti sieno di
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2244 detta famosa quantità. </s>
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2245 </p>
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2246 <p type="main">
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2247 <s>Gli angoli, oltre acciò collocati dentro a’ cerchi quantunque
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2248 <lb></lb>//<lb></lb>disuguali, con la quantità loro mostrano le portioni simili,
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2249 come insegna da Euclide nella .11. def. del .3°.. L’angolo retto posto nel
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2250 cerchio ci dà inditio del mezzo cerchio, come si vede nella .10. def. del
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2251 3°. Questi son tutti effetti ed utilità che gli angoli producono nella
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2252 Geometria, ed altre anchora de’ quali non si favellerà per hora in questo
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2253 luogo per non allongar troppo il discorso; ma se ne lassarà la cura a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2254 qualunque osservarà i rimanenti libri di Euclide bastandoci solo haver
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2255 accennato in parte l’uso degli angoli nella Geometria.</s>
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2256 </p>
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2257 <p type="main">
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2258 <s> </s>
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2259 </p>
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2260 </chap>
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2261 <chap>
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2262 <p type="head">
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2263 <s>Nell’Astronomia</s>
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2264 </p>
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2265 <p type="head">
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2266 <s>Cap. 12</s>
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2267 </p>
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2268 <p type="main">
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2269 <s>Se ‘l cielo dee sempre muovarsi e con facil giramento intorno alla Terra, e
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2270 se ‘l suo girare non dee esser interrotto da intervallo alcuno di
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2271 piccolissima quiete, né proceder saltando, necessariamente li conviene la
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2272 figura in tutto ed ogni maniera d’angoli privata. </s>
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2273 <s>E se di questa maniera si fa obbietto degli Astronomi per qual cagione nelle
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2274 contemplationi loro si ricercano le considerationi e gli usi delle figure
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2275 angolari e degli angoli stessi? Tolomeo nel 3° cap. del primo lib.
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2276 dell’Almagesto afferma che alla struttura degli stromenti del movimento
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2277 celeste non conviene altra figura che la Sferica; perciochè ‘l movimento
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2278 delle cose celesti, non essendo impedito da alcuna cosa avviene che via più
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2279 d’ogni altra cosa facilissimamente giri. </s>
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2280 <s>E di tutte le figure dico nelle superficie la circolare: e ne’ solidi la
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2281 sferica facilmente si muove. </s>
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2282 <s>Ma che occorre dubbitare della natural figura del Cielo; se gli Astronomi
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2283 oltre ad essa con la varia consideratione loro e per cagion di pruovare
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2284 alcuni effetti con dimostrationi matematiche, non solo hanno diviso ‘l Cielo
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2285 in molti cerchi; fondati però nelle diversità de’ movi <pb pagenum="folios 40v-41r"></pb>menti, come si può veder appo ‘l Clavio nella
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2286 Sfera, e vi sia collocato anchora altri cerchij nuovi, col mezzo de’ quali
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2287 mostrano la diversità dell’apparenza de’ pianeti; ma ancho vi hanno formato
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2288 diverse figure angolari, e diversi intersegamenti d’archi e per conseguenza
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2289 anchora molte e diverse maniere d’angoli per dimostrare alcune cose, che con
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2290 ragioni naturali pruovar non si potevano. </s>
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2291 <s>Di modo che supposta la figura natutale de’ Cieli gli Astronomi forzati a
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2292 dimostrar i varij effetti di essi, e non havendo altre dimostrationi, che le
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2293 matematiche, e non convenendogli usar quelle de’ Filosofi naturali, oltre
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2294 all’altre cose costituirono nelle Celesti Sfere diversi cerchij posti in
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2295 diversissimi modi, ed in guisa che tagliandosi insieme, formano angoli tal
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2296 hora retti, tal hora ottusi e tal hora acuti; onde risultano ancho diverse
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2297 specie di figure e varie portioni d’archi. </s>
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2298 <s>Né questa fu solamente pura immaginatione loro, ma imaginatione ritratta
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2299 dalla osservation de’ corpi celesti i quali fra loro muovendosi diversamente
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2300 e secondo diverse positure, formano varij giri che tagliandosi variamente
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2301 insieme, costituiscono diverse specie d’angoli e di figure. </s>
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2302 <s>Onde non è maraviglia se Proclo nel 2° lib. d’Euclide, nel cap. 11° dica che
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2303 agli Dei delle progressioni e del movimento, donatori delle varietà delle
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2304 potenze sì come dano gli angoli ottusi e gli acuti. </s>
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2305 <s>E che sono altro questi Dei che l’Intelligenze, che assistono a’ corpi
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2306 Celesti per dargli ‘l movimento sì come è commun parere de’ Peripatetici?
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2307 Anzi sono l’anime de’ Cieli, come è oppinione di essi e de’ Platonici
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2308 anchora, sì come afferma Marsilio Ficino nella Teologia di
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2309 <lb></lb>//<lb></lb>Platone e ‘l Cardinal Bessarione nelle calunnie dell’istesso
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2310 cap. 10° del 2° lib. </s>
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2311 <s>Queste secondo l’oppinion di Guglielmo Postello nel lib. dell’Origine della
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2312 Toscana etc. sono i Genij stessi così appellati; perciochè sovrastano alla
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2313 generatione in tutti que’ luoghi a quali assistono: e conservono tutte le
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2314 generationi degli animali e delle piante nelle nature loro determinate a una
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2315 specie, ad un luogo e ad una maniera di gente. </s>
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2316 <s>Onde chiunque è nato in Grecia, grecheggia, o ‘n Italia, italianeggia, o ‘n
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2317 Francia, franceseggia per natura e ciò si potrebbe confermare col detto di
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2318 Marsilio Ficino sopra Plotino nella En. seconda. </s>
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2319 <s>Egli dice in quel luogo, che i Celesti Dei provedono ciascuna cosa agl’
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2320 inferiori, overo, con l’autorità del Ficino sopra Protagora e nel 12° cap.
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2321 del Timeo di Platone, diciamo che gli Dei suddetti non sieno altro che le
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2322 stelle, le quali col movimento loro formano varij cerchij che intersegandosi
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2323 insieme producono angoli diversi, come si è detto. </s>
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2324 <s>E perché si chiamano donatori delle potenze delle varietà, se non perché
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2325 infondono nelle cose inferiori varie dispositioni e diverse ragioni di forme
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2326 sostantiali e son cagione in noi di quasi innumerabili inclinationi? Tutti
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2327 effetti maravigliosi cagionati da’ Cieli in queste basse cose. </s>
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2328 <s>Ma poniamo da parte queste filosofiche speculationi e torniamo al proposito
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2329 nostro. </s>
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2330 <s>Gli angoli adunque delle figure celesti che appo Jacomo Carpentario, nel cap.
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diff changeset
2331 11° sopra Alcinoo, dagli Oracoli son chiamati congiognimenti e nodi;
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2332 perciochè sono immagini delle unioni ristregnementi e delle divine
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2333 congiuntioni per le quali quelle cose che per natura loro son separate
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2334 scambievolmente si accos <pb pagenum="folios 41v-42r"></pb>tano per molti
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2335 rispetti son cagione di varij giovamenti nell’Astronomia. </s>
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2336 <s>Per gli oracoli si intende la ragion del pronosticare; e le divine
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2337 congiuntioni; sono de’ corpi celesti e così le figure. </s>
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2338 <s>Perciochè i cerchij della Sfera maggiori e minori, secondo la positura loro,
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2339 o retta od obliqua, s’intersegano scambievolmente, e ‘l tagliamento loro si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2340 determina e si manifesta dalla costitution degli angoli retti o non retti;
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2341 perciochè quando le uguali circonferenze o le disuguali di maniera si
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2342 tagliano nella superficie della Sfera, che intorno al ponto del tagliamento
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2343 commune si cagionino angoli uguali il che non accade se non ne’ cerchij
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2344 grandi: o solamente da una banda due angoli collaterali, e gli altri due
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2345 rimanenti dall’altra son fatti uguali; la qual cosa non suole accader se non
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2346 nelle circonferenze uguali, allhora due cerchij si tagliano ad angoli retti
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2347 e così per opposito. </s>
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2348 <s>E questo si vede espressamente del tagliamento dell’Orizzonte fatto
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2349 dall’Equinottiale. </s>
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2350 <s>Perciochè quelli che hanno la Sfera retta vedono l’Equinottiale esser
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2351 tagliato dall’Orizzonte ad angoli retti: e que’ che l’hanno obliqua ‘l
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2352 vedono tagliato dall’Orizzonte ad angoli disuguali e obliqui; perciochè in
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2353 questo caso discernono solamente gli angoli opposti uguali, o veruno esser
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2354 uguale; onde i cerchij e le circonferenze loro obliquamente si segano
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2355 insieme e l’una declina dall’altra. </s>
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2356 <s>Oltre acciò nel nascimento degli angoli nella Sfera si fa la division
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2357 dell’Eclittica e dell’Equatore in quattro quarte, fatta dall’Orizzonte e dal
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2358 Cerchio Meridiano, come si può ritrarre dal 10° cap. del 2° dell’Almagesto
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2359 di Tolomeo. </s>
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diff changeset
2360 <s>E ‘punti termi <lb></lb>//<lb></lb>nanti le dette quarte sogliono esser chiamati
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diff changeset
2361 cardini ed angoli; poiché il tagliamento di queste quarte non si può eseguir
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2362 senza formar gli angoli nella sfera. </s>
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diff changeset
2363 <s>Neancho senza formar’ angoli il Zodiaco, uno de’ cerchij maggiori sega
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diff changeset
2364 l’Equinottiale in due parti uguali, ma obliquamente. </s>
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diff changeset
2365 <s>Così anchora i coluri mentre passano pe’ Poli del Mondo non tagliano i
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diff changeset
2366 cerchij maggiori e minori senza formare angoli retti. </s>
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diff changeset
2367 <s>Si aggiogne che ‘l Zodiaco stesso non si divide in dodici Segni senza la
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diff changeset
2368 formation degli angoli retti; onde da alcuni sono stati detti quadrati o
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diff changeset
2369 quadrangolari: ansi ciascun segno non per altro si appella piramide
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diff changeset
2370 quadrilatera, se non perché essendo esso la base, ha quattro lati sorgenti
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diff changeset
2371 da quattro angoli che terminano nella terra e formano angolo acuto. </s>
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diff changeset
2372 <s>E tutti questi tagliamenti, benchè si mostrino nella Sfera materiale, con
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diff changeset
2373 tutto ciò s’immaginano ancho nella celeste; poiché la materiale si forma per
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2374 la intelligenza della celeste. </s>
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diff changeset
2375 <s>Oltre acciò ‘l nascimento e l’uso degli angoli è giovevole espressamente alla
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diff changeset
2376 figura della dispositione del Cielo. </s>
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diff changeset
2377 <s>Perciochè (sì come si ritrahe da Gio. da Monte Regio nella disputa contra ‘l
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parents:
diff changeset
2378 Cremonese) il cielo si divide in tre modi. </s>
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diff changeset
2379 <s>Il primo si è quello che si fa da sei cerchij grandi, che si tagliano insieme
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diff changeset
2380 sopra i punti opposti, o sieno poli del Zodiaco o poli del Mondo, o ‘due
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diff changeset
2381 punti dell’Orizzonte e del Meridiano. </s>
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diff changeset
2382 <s>Il diametro de’ quali si parte dal Settentrione, e arriva al Mezzogiorno, ed
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diff changeset
2383 essi passano sopra dodici tagli, immaginati nell’Equinottiale, a’ quali
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diff changeset
2384 corrispondono XII parti del Zodiaco, dette immagini, case habi <pb pagenum="folios 42v-44r"></pb>tacoli, torri, borghi, hospitij, ricettacoli,
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diff changeset
2385 luoghi, immagini e castelli: e dividono qualunque cerchio per due punti
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diff changeset
2386 opposti, e tutto ‘l Cielo e tutta la macchina del Mondo in dodici parti; di
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diff changeset
2387 maniera che ciò che è nel Mondo si ritruovi in alcuna di queste parti dette
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2388 case. </s>
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diff changeset
2389 <s>Tal che ‘l primo modo si vedrà descrivendo due circonferenze concentriche, e
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diff changeset
2390 dividendole in XII parti uguali: overo descrivendo la Sfera e costituito
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diff changeset
2391 l’Equinottiale, i poli e l’asse: e formandovi sei cerchij che si taglino ne’
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parents:
diff changeset
2392 poli e dividino in XII parti il cerchio Equinottiale. </s>
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diff changeset
2393 <s>Il 2° modo risolve tutta la disposition del Cielo in un quadrato, dividendo
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diff changeset
2394 ciascun de’ suoi lati in tre parti uguali; onde congiognendo tutti i punti
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2395 della divisione tutta l’area del quadrato vien divisa in nove quadrati
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2396 uguali e divisi i quadrati che son negli angoli della figura col mezzo del
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diff changeset
2397 diametro tosto apparisce la division di tutto ‘l cielo in XII parti intorno
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2398 al quadrato di mezzo. </s>
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diff changeset
2399 <s>Quattro delle quali opposte per diritto formano una croce, e si appellano
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2400 cardini overo angoli, cioè la I. la IIII. la VII. e la X. e le otto parti
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2401 rimanenti sono otto triangoli, che raffigurano la II. la III. la V. la VI.
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2402 l’VIII. la IX. l’XI. e la XII. parte del Cielo. </s>
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diff changeset
2403 <s>Il terzo modo si è quando si formano due quadrati, uno dentro all’altro, un
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2404 maggiore e l’altro minore, ed un altro quadrato, i cui lati toccando gli
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2405 angoli del quadrato minore, formino gli angoli nel mezzo de’ lati del
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2406 maggiore, dividendosi ciascun lato in due parti uguali: e tirati i diametri
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2407 del maggiore, apparisce tutta la <lb></lb>//<lb></lb>figura divisa in XII triangoli
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2408 posti intorno al quadrato minore, i quali corrispondono alle XII parti del
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2409 Cielo. </s>
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2410 <s>Di modo che qualunque sia di queste divisioni, che rappresenti ‘n che maniera
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2411 sia disposta la figura del Cielo non si può formar senza cagionar angoli
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2412 diversi; poiché quindi apparisce risolversi il Cielo in XII. parti
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2413 raffigurate da altretante figure angolari. </s>
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2414 <s>E per qual cagione Marsilio Ficino nel cap. 12 sopra ‘l Timeo di Paltone
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2415 ragionando de’ corpi celesti, disse che in maniera concorreno che insieme
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2416 nascono o veramente quasi per linea perpendicolare si congiongano, come si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2417 può vedere ancho appo Girolamo Fracastoro nel cap. 21° dell’ Homocentrica,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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2418 se non perde dalla cognition degli angoli haveva già ritratta la notitia di
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2419 tal cagionamento? Che non poteva la congiuntione in modo alcuno farsi per
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diff changeset
2420 linea perpendicolare senza gli angoli perciochè (sì come è manifesto per la
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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2421 Geometria) non è linea alcuna che si possa chiamar perpendicolare senza
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2422 formar angoli da ogni banda, cadendo sopra altra linea. </s>
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parents:
diff changeset
2423 <s>E se riguardiamo alla Sfera materiale, la quale corrisponde alla naturale,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2424 noi vedremo chiarissimamente, in fra ‘tagliamenti de’ cerchij maggiori e de’
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parents:
diff changeset
2425 minori, adoperarsi l’uso degli angoli. </s>
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parents:
diff changeset
2426 <s>Perciochè, osservando noi i cerchi maggiori vediamo il Meridiano e
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parents:
diff changeset
2427 l’Orizzonte tagliarsi ad angoli retti, stando la Sfera retta, ne’ poli del
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2428 Mondo; e nel ponto Verticale appellato Zenit, cadente sopra ‘l nostro capo:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2429 e in uno stesso tempo amendue tagliare l’Equinottiale pur ancho ad angoli
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2430 retti. </s>
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diff changeset
2431 <s>Ma considerata la Sfera <pb pagenum="folios 43v-44r"></pb>obliqua, l’Orizzonte
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parents:
diff changeset
2432 col Meridiano si tagliano pure ancho ad angoli retti; ma fuor del polo del
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2433 Mondo; perché a que’ che hanno la Sfera torta il polo s’inalza sopra
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parents:
diff changeset
2434 l’Orizzonte, e ‘n questo caso l’Equinottiale è segato dall’Orizzonte ad
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parents:
diff changeset
2435 angoli disuguali e obliqui, cioè ottusi, ed acuti. </s>
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parents:
diff changeset
2436 <s>E se rimiriamo ‘l cerchio Equinottiale, vedremo che taglia formando angoli
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2437 retti sferici il primo mobile; onde è dela sua cintura come riferisce M°
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parents:
diff changeset
2438 Mauro Fiorentino Servita nelle Notationi della Sfera. </s>
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diff changeset
2439 <s>Né può esserne cintura senza formare angoli; perciochè, mentre ‘l cegne il
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parents:
diff changeset
2440 taglia, segando ancho ‘l Zodiaco ad angoli disuguali. </s>
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diff changeset
2441 <s>Onde possiamo concludere che dalla qualità degli angoli si conosca la
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parents:
diff changeset
2442 positura della Sfera, cioè se sia o retta o obliqua e così ancho la maniera
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2443 de’ tagliamenti fatti da’ cerchij maggiori. </s>
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parents:
diff changeset
2444 <s>Oltre acciò se osserviamo i cerchij Coluri, vedremo che si tagliano fra loro
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parents:
diff changeset
2445 ad angoli retti ne’poli Mondo: e mentre tagliano ad angoli disuguali il
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diff changeset
2446 Zodiaco, passando pe’ poli Solstiali, cioè del Cancro e del Capricorno, la
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parents:
diff changeset
2447 State e l’Inverno, determinare i due Solstitij e due Equinotij. </s>
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2448 <s>E questi punti non si possan costituir senza la formation degli angoli, che
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2449 necessariamente procede da’ communi tagliamenti de’ cerchij. </s>
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2450 <s>I medesimi Coluri tagliono ad angoli retti i due cerchij minori chiamati
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2451 Tropici, del Cancro e del Capricorno; onde il Sole si rivolge e si accosta
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2452 di State al nostro Zenit, e di Verno a quello degli Antipodi. </s>
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2453 <s>Nel medesimo modo tagliono l’Orizzonte e ‘l Meridiano cioè nella Sfera retta,
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2454 ma nell’obliqua ad an <lb></lb>//<lb></lb>goli disuguali. </s>
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2455 <s>E quindi ancora si può imparare a discerner la Sfera diritta dalla torta. </s>
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2456 <s>E questo basti per dimostrar l’utilità degli angoli nel commun segamento de’
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2457 cerchij maggiori della Sfera. </s>
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2458 <s>Hora se ci voltiamo a’ minori, i quali sono il cerchio Artico e Antartico, il
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2459 Tropico del Cancro e del Capricorno, vedremo espressamente che tutti
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2460 tagliano ‘l Meridiano l’Orizzonte retto e ‘Coluri ad angoli retti: e così
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diff changeset
2461 l’asse del Mondo; ma l’Orizzonte obliquo e l’asse del Zodiaco ad angoli
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2462 obliqui e disuguali. </s>
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2463 <s>Fra ‘ due Tropici muovendosi ‘l Sole, forma 182 spire, overo cerchij involuti
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2464 i quali raddoppiati nel suo ritorno e computato due volte l’Equinottiale sul
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2465 quale gira due volte si formano giorni 365 in tutto l’anno, e quasi 6 hore e
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2466 21. M. meno, che sono la centesima parte d’un giorno. </s>
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2467 <s>E le dette spire tagliando l’Eclittica, il Meridiano, l’Orizzonte retto e
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2468 obliquo e ‘Coluri e gli assi del Mondo e del Zodiaco, formano varie specie
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2469 d’angoli. </s>
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2470 <s>Di modo che le dette spire tante sieno quanti i tagliamenti e ‘tagliamenti
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2471 quanti gli angoli e quanti gli angoli o tagliamenti o le spire, tanti sieno
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2472 i giorni. </s>
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2473 <s>Queste dall’Orizzonte nella Sfera diritta son tagliate in parti uguali,
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2474 mentre egli passa sopra ‘ Poli del Mondo; onde risultano i giorni uguali. </s>
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2475 <s>Ma essendo divise da esso nella Sfera torta, stando esso obliquamente, si
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2476 cagiona la disuguaglianza e la diversità de’ giorni e delle notti in diverse
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2477 parti del Mondo. </s>
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2478 <s>Di maniera che queste divisioni non si possendo eseguir senza formare angoli
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2479 e giorni cagionandosi uguali o disuguali col mezzo di esse seguirà che da
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2480 <pb pagenum="folios 44v-45r"></pb>gli angoli anchora si cagionino l’ugualità
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2481 e le disuguaglianze de’ giorni. </s>
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2482 <s>E se è lecito dalle celesti Sfere trapassar scendendo al globo della terra,
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2483 riputata consorte del Cielo da Leone Helode, nel 2° Dialogo d’Amore, potremo
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2484 dire che ‘quattro cerchij che la dividono in cinque zone, ciò sono i due
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2485 cerchietti, Artico e Antartico: e’ due Tropici del Cancro e del Capricorno;
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2486 formano nella circonferenza della terra angoli retti sferali, nella stessa
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2487 guisa che ne’ cerchij Celesti per la Commune proportione che è fra ‘
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2488 cerchijj che sono intorno al medesimo centro. </s>
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2489 <s>Il che avviene come nel divider in quarte o in quadrati ‘l Cielo e la Terra;
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2490 là dove essendo il quadrante della Terra al quadrante del Cielo simile e
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2491 proportionale, e l’angolo che si fa nel quadrante di amendue havendo
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2492 proportione, ansi essendo uguale, segue che ‘l cerchio della Terra contenuto
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2493 da quello del Cielo sia proportionato e simile al cerchio del Cielo,suo
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2494 continente, sì come si dimostra dal Glareano nel 12° cap. della Geografia,
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2495 di modo che e la Terra el Cielo parimente si divide in quattro parti. </s>
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2496 <s>Così i detti quattro cerchij che dividono il cielo in cinque parti dividono
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2497 ancho la Terra in altretante, contenute sotto le dette portioni del Cielo, e
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2498 a esse proportionali e simili. </s>
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2499 <s>E come nella circonferenza celeste i detti cerchij formano angoli retti
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2500 sferali, così nella circonferenza terrena. </s>
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2501 <s>Ma poiché di sopra si è fatta mentione del Zenit, non si dee tralassare, che
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2502 anch’esso, essendo un punto collocato nella Sfera diritta, ne poli del
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2503 Mondo, da esso cade una linea perpendicolare sopra ‘l <lb></lb>//<lb></lb>nostro
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2504 capo, overo in quella parte della superficie della Terra dove terminano i
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2505 piedi, onde essendo il nostro capo di figura sferica, ma alquanto più longa
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2506 che larga, sì come afferma Galeno nel cap. 17° del 9° lib. </s>
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2507 <s>Dell’uso delle parti e nel 1° cap. dove favella dell’ossa, e per la sperienza
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2508 si conferma: e la detta portione essendo parte della circonferenza della
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2509 terra, non può non produr angoli misti nello stesso contatto, sì come ancho
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2510 partendosi del concavo della Sfera. </s>
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2511 <s>Oltre acciò le dimostrationi sferiche, con le quali si pruovano le grandezze
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2512 degli archi massimi che si descrivono sopra i poli dell’equinottiale, i
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2513 quali son collocati fra esso e l’eclittica, mentre in esse si fa la
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2514 conferenza delle proportioni composte delle linee, non si fanno senza
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2515 formare angoli nelle descrittioni loro; poiché si ordinano producendo linee,
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2516 delle quali si compongano triangoli con lati proportionali, sì come si vede
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2517 appresso Tolomeo nel cap. 12° del primo lib. dell’Almagesto, là dove si
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2518 mostra la proportion d’una linea maggiore ad una minore, che è parte di
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2519 essa, esser composta delle proportioni delle linee minori alle parti loro,
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2520 la qual cosa concludendosi nelle linee si conlcuderà anchora negli angoli e
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2521 negli spatij triangolari contenuti da esse. </s>
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2522 <s>Il che è quasi lo stesso che quel che afferma de’ triangoli Euclide nella
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2523 quarta del primo, che non vi è altra differenza che dall’esser uguale
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2524 all’esser proportionale. </s>
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2525 <s>Come se dicessimo: se due triangoli hanno due lati proportionali a due lati,
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2526 e un angolo proportionale a un angolo contenuto da linee rette
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2527 proportionali; haveranno ancho la base proportionale alla <pb pagenum="folios 45v-46r"></pb>base e ‘l triangolo al triangolo e gli altri
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2528 angoli agli altri angoli, e l’uno all’altro, a quali soggiacciono lati
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2529 proportionali. </s>
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2530 <s>Di modo che quindi vediamo, che ‘l medesimo effetto, che si cagiona da’ lati
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2531 proportionali de’ triangoli, si cagiona anchora dagli angoli, cioè ‘l far le
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2532 basi, i triangoli e gli altri angoli proportionali; onde segue la
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2533 proportionalità degli archi a’ quali son sottoposti per l’.XI. def. del 3°
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2534 d’Euclide e per la XII del Commandino e ciò vi si vede espressamente; poiché
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2535 si fa comparatione fra le corde degli archi proposti e si mostra tale esser
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2536 la proportione in fra gli archi, che è in fra le corde, le quali son linee
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2537 rette e fanno angoli nelle circonferenze e ne’ tagliamenti loro. </s>
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2538 <s>Ma ‘l tutto si dichiara da Tolomeo con sette dimostrationi nel med. lib. </s>
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2539 <s>Ma nel cap. 13, per gli mostrar la declinatione di qual si voglia punto
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2540 dell’eclittica, cerca la proportione d’un arco minore e truovata la doppia
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2541 proportione della corda d’un arco maggiore alla doppia proportione della
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2542 corda d’una parte di lui, cioè d’un arco minore, esser composta di
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2543 proportioni delle corde degli archi minori e del doppio della misura loro;
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2544 procede mostrando la proportion degli archi in fra loro, facendo sempre
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2545 comparation degli archi agli archi e delle corde alle corde; onde con
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2546 scemamento di gradi si perviene finalmente all’arco della declinatione
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2547 dell’eclittica, e al termine di essa. </s>
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2548 <s>E ciò si vede in una descrittion del cerchio Meridiano, diviso da due mezzi
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2549 cerchij ad angoli retti sferici che si tagliano nel centro di esso ad angoli
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2550 disuguali, dove costituiscono il ponto dell’Equinottio dell’Inverno. </s>
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2551 <s>Uno de’ <lb></lb>//<lb></lb>quali cerchij è obliquo ed è la stessa eclittica, e
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2552 l’altro diritto, ed è l’Equatore: ed amendue terminano in due punti della
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2553 circonferenza del Meridiano, là dove sono i due Tropici del Cancro e del
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2554 Capricorno, e a man destra nella medesima circonferenza è ‘l polo del
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2555 Zodiaco da cui nasce un arco che taglia l’eclittica e l’equinottiale ad
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2556 angoli disuguali e ne’ tagliamenti costituisce la portion d’un arco minore,
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2557 dove si vede collocarsi il punto della declinatione della eclittica. </s>
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2558 <s>Di maniera che in questa descrittione dimostrando Tolomeo la declination di
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2559 qualunque punto dell’eclittica; ci fa conoscere che i detti cerchij fra loro
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2560 si tagliano proportionalmente; onde seguono diversi archi proportionali, e
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2561 così diverse corde altresì proportionali: e quindi ancho quattro triangoli
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2562 proportionali; onde avviene che faccia la circonferenza delle proportioni
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2563 loro. </s>
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2564 <s>Ne’ triangoli, ne’ lati di essi, i quali sono specie d’archi, possono esser
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2565 proportionali, se gli angoli anchora non son proportionali. </s>
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2566 <s>Il medesimo si può dire che si faccia nella seconda dimostratione, dove si
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2567 mostra, che ‘l tagliamento retto di qualunque arco dell’eclittica si
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2568 comincia dalla settion di essa e dell’equatore. </s>
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2569 <s>Oltre acciò (come si ritrahe dal 10° cap. del 2° dell’Almagesto) per virtù
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2570 degli angoli si viene in cognition de’ cerchij massimi della Sfera Celeste,
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2571 che gli contengono, mentre da essi sopra ‘poli si prende lo spatio commune:
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2572 e della quarta parte del cerchio costituita dall’arco infraposto alle parti
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2573 che fanno angolo. </s>
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2574 <s>Di più dalla misura e proportion dell’angolo sottoposto all’arco detto, e <pb pagenum="folios 46v-47r"></pb>paragonata a quattro angoli retti possiamo
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2575 conoscere la proportion dell’arco interposto a tutto ‘l cerchio. </s>
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2576 <s>Onde se l’angolo formato nel tagliamento del cerchio sarà di 90 parti, di
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2577 tante sarà l’arco, ed essendo nel taglio quattro angoli retti de’ quali
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2578 ciascuno sia 90 parti, seguirà che tutti insieme sieno 360: ed essendo
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2579 quattro archi infraposti fra ‘termini degli angoli detti corrispondenti agli
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2580 angoli retti, ciascuno per sé sarà 90 parti, e tutti insieme 360. E per
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2581 seguir più avanti recitano la sentenza di Tolomeo. </s>
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2582 <s>Adunque degli angoli che si formano secondo ‘l cerchio obliquo, quegli a
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2583 questa speculatione (astronomica) grandemente son giovevoli che son compresi
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2584 dal tagliamento di esso, del Meridiano e dell’Orizzonte in qual si voglia
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2585 sito. </s>
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2586 <s>Così anchora que’ che son contenuti dalla settion di esso e dal cerchio
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2587 massimo descritto sopra ‘poli dell’Orizzonte. </s>
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2588 <s>E con quest’angoli insieme si dimostrano gli archi, che s’interpongono fra ‘l
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2589 tagliamento el polo dell’orizzonte, cioè del ponto verticale, overo Zenit;
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2590 perciochè ciascuna di queste cose dimostrata conferisce molto alla stessa
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2591 speculatione, e a tutto quel che si cerca nelle diversità degli aspetti
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2592 della Luna. </s>
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2593 <s>E questo basti haver riferito per dimostrar l’utilità degli angoli. </s>
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2594 <s>Chi ne desidera le dimostrationi ricorra a Tolomeo nel medesimo luogo citato. </s>
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2595 <s>Ma poiché si sono accennati gli aspetti, questo solo appartenente a essi
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2596 aggiognerò, che la diversità di essi, la quale apparisce nelle quadrature,
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2597 non si può far senza <lb></lb>//<lb></lb>formare angoli. </s>
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2598 <s>Perciochè (sì come si vede appresso Girolamo Fracastoro nel cap. 21°
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2599 dell’Omocentrica) le quadrature, l’oppositioni, e le congiuntioni si formano
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2600 in due linee, una delle quali si appella asse e l’altra antasse: e amendue
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2601 tagliando i deferenti, e l’eccentrico, mentre si segano insieme ad angoli
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2602 retti costituiscono i punti dell’oppositione e della congiuntione, ne’ quali
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2603 il Pianeta ritruovandosi apparisce due volte vicino e due lontano. </s>
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2604 <s>Di maniera che nella congiuntione e nell’oppositione, la Luna essendo sempre
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2605 nell’asse; e nelle quadrature, essendo nell’antasse, dimostra la varietà
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2606 degli aspetti per mezzo degli angoli costituiti da’ tagliamenti di due linee
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2607 rette. </s>
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2608 <s>Questi aspetti della Luna nelle quadrature non si cagionano ne’ cerchij
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2609 eccentrici deferenti (detti Draghi, i cui tagliamenti ad angoli retti
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2610 sferali son chiamati il capo e la coda del Drago) senza gli angoli perciochè
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2611 se riguardiamo all’asse, che determina la maggior longhezza e distanza della
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2612 Luna dal centro del Mondo: e dell’antasse, che dimostra la vicinanza,
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2613 vedremo che ‘n fra loro segandosi formano per ogni verso angoli retti: e
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2614 così anchora se riguardiamo a’ termini nel taglio dell’ovato vi scorgeremo
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2615 quattro angoli retti, in quanto all’esser loro; ma in quanto all’apparenza
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2616 due ottusi e due acuti, dove si termina tutto l’ovato, cioè immaginandoci
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2617 che ‘ due punti della congiuntione e dell’oppositione e degli accostamenti
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2618 si congionghino e in questa guisa si vedrà formata la figura quadrata, che
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2619 in <pb pagenum="folios 47v-48r"></pb>apparenza si mostra un rombo dentro l’ovato
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2620 prodotto dal ristregnimento de’ deferenti. </s>
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2621 <s>Ma si avvertisca che di questo cerchio, che dal Fracastoro è appellato ovato,
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2622 avviene come degli altri cerchij; che per la varia positura loro, opposti
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2623 alla veduta nostra, si mostrano di figura difettuosa, come è l’ellipse o
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2624 l’ovato, come si potrebbe pruovare per ragion di Prospettiva; che questo
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2625 cerchio non per altro è detto cerchio ovato, se non che in apparenza si
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2626 mostra ovato, come è ‘l taglio obliquo della piramide o del cilindro; ma
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2627 nell’esser suo è cerchio perfetto, come gli altri della Sfera, che opposti
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2628 all’occhio, dimostrano così fatta apparenza. </s>
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2629 <s>Gli angoli finalmente servono negli aspetti triangolari, quadrati, e sestili,
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2630 ne’ quali si ritruovano le proportioni armoniche, cioè tripla, sesquialtera,
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2631 sesquitertia, come nelle voci si formano le consonanze: Diapente,
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2632 Diatesseron, come riferisce Cesare Cesariano nel cap. primo del 1° di
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2633 Vitruvio, ritrahendolo dal 2° del Quadripartito di Tolomeo. </s>
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2634 <s>E questo basti in quanto a questa parte dell’uso degli angoli
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2635 nell’Astronomia; che per hora non intendiamo dirne altro, rimettendoci in
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2636 tutto a’ diligenti osservatori delle cose Celesti.</s>
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2637 </p>
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2638 <p type="main">
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2639 <s></s>
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2640 </p>
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2641 </chap>
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2642 <chap>
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2643 <p type="head">
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2644 <s>Nella Prospettiva</s>
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2645 </p>
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2646 <p type="head">
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2647 <s>Cap. 13</s>
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2648 </p>
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2649 <p type="main">
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2650 <s>Non è dubbio alcuno che se consideraremo bene tutte le parti della
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2651 Prospettiva, o appartenghino all’ombre, dico alle scene, o a reflettimenti
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2652 de’ raggi del Sole, o agli specchi, overo alle linee e raggi visuali, noi vi
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2653 conosceremo tutta la forza delle pratiche e tutta l’efficacia delle
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2654 dimos<lb></lb>//<lb></lb>trationi nascen dalla diversa aplicatione degli angoli, la
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2655 qual cosa prenderemo a dimostrare. </s>
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2656 <s>Ma perché ci si offerisce la parte Ottica, la quale appartiene al modo con
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2657 cui si eseguisce la visione, però prima ragionaremo di questa, ma avanti si
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2658 dee supporre che Euclide, essendo Filosofo Platonico, nella sua Prospettiva
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2659 dimostra esser dell’oppinion del suo maestro Platone, in quanto appartiene
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2660 al modo col quale si fa la visione ed è che si faccia col mezzo di raggi
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2661 visuali, ch’escono dall’occhio e vanno a truovarsi l’obbietto. </s>
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2662 <s>Il che da lui si fece chiaro quando nella prima suppositione della
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2663 Prospettiva disse i raggi visuali uscir dall’occhio e andar a truovar
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2664 l’obbietto. </s>
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2665 <s>Della quale oppinione sono stati i suoi seguaci e commentatori, contro a’
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2666 quali si è mostrato Giovanni Arcivescovo di Cantauria nella prima parte
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2667 della sua Prospettiva comune nella 44 e 45 conclusione, là dove disse: i
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2668 matematici indarno affermare che la visione si faccia pe’ raggi che escono
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2669 dall’occhio: ed esser cosa impossibile che i detti raggi uscendo dall’occhio
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2670 alla cosa veduta sieno bastevoli a formar la visione *(e ‘l Cavalier Lorenzo
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2671 Sirigatti nel 3° cap. del primo libro della Pratica di Prospettiva) [nota in
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2672 margine]. E benchè l’oppinion d’Euclide non sia conforme alla dottrina
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2673 Peripatetica, contuttociò trattandosi di Prospettiva insieme con esso non si
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2674 disconverrà seguirla; quantunque secondo l’oppinion Aristotile non si mostri
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2675 sconvenevole ragionarne; poiché, con altro adattamento di raggi visuali
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2676 affermaremo farsi la visione, discendo che invece di uscir dal centro
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2677 dell’occhio e andar afferir nell’obbietto, scaturischino dall’estremi<pb pagenum="folios 48v-49r"></pb>tà dell’obbietto e si terminano tornandosi nel
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2678 centro dell’occhio, acciochè la specie della cosa veduta, passando pe’ raggi
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2679 visuali, arrivi all’occhio, onde si faccia la perfetta visione. </s>
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2680 <s>Ma per dar principio ad eseguir ciò che si è proposto si dee dire che l’uso
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2681 degli Angoli nella Prospettiva Ottica si scuopre in ogni risguardamento
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2682 d’obbietto ed in ciascuna dimostratione degli effetti che secondo diverse
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2683 apparenze e secondo varia posizione si mostrano diversi. </s>
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2684 <s>Non è dubbio alcuno che seguendo noi in questo luogo il parer de’ Platonici
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2685 nel producimento della visione ci faccia bisogno affermare con Euclide che
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2686 non possiamo veder obbietto alcuno se i raggi visuali non escono dall’occhio
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2687 per andar dirittamente a trovar l’obbietto e se non hanno nella parte più
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2688 lontana intervallo, sicome si vede appresso la prima supposizione della
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2689 Prospettiva. </s>
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2690 <s>E mentre i detti raggi, che non sono altro che linee rette escono
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2691 dall’occhio, quanto più si allontanano da esso e si avvicinano alla cosa
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2692 veduta, tanto più si discostano; onde necessariamente formano angolo; ansi
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2693 non possan haver tra loro intervallo senza formare angolo, che uscendo dal
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2694 centro dell’humor cristallino ed essendo l’obbietto molte volte maggiore
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2695 dell’occhio e dovendo terminarsi i raggi nella estremità di esso, acciò si
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2696 faccia la visione, non possan costituir intervallo secondo la misura
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2697 dell’obbietto: e questo non si può fare se prima no si forma l’angolo nel
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2698 centro dell’occhio. </s>
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2699 <s>E però Euclide, nella seconda supposizione disse la figura formata da’ raggi
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2700 visuali esser un conio la cui ponta è nell’occhio e la base nella
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2701 <lb></lb>//<lb></lb>stremità della cosa veduta (al mezzo dentro a questo conio si
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2702 forma la visione, quasi che la virtù apprensiva dell’occhio si muova su pe’
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2703 raggi visuali per accostarsi all’obbietto. </s>
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2704 <s>Ma se questo conio è una piramide ritonda come vuole Euclide nella def. 16
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2705 dell’XI e come piace a Vitellione nel 4° libro) e se essendosi tonda ha per
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2706 base ‘l cerchio; onde è che si fa la visione anchora quando l’obbietto non è
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2707 di figura circolare o sferica? Non conviene affermare che sotto ‘l nome del
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2708 Conio si contenga ogni specie di piramide, come alcuni erroneamente hanno
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2709 pensato; poiché dall’esser del conio sono al tutto lontane le figure
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2710 angolari. </s>
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2711 <s>Né si dee dire che ‘l conio, benchè ritondo possa abbracciar le figure
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2712 angolari; perciochè la figura ritonda non può tutta ugualmente accostarsi
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2713 all&#039;’ngolare, non vi essendo commune proportione alcuna; perciochè la
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2714 tondezza del conio o escede la figura angolare o ‘l conio da essa è
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diff changeset
2715 superato.Neancho è ragionevole ‘l pensar che ‘l conio si adatti in maniera
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diff changeset
2716 ad ogni figura angolare, che di tondo si faccia angulare; perciochè, tosto
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diff changeset
2717 che diviene angolare, non è più conio; perciochè, acciochè si faccia
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diff changeset
2718 angolare bisogna che a ciascuno angolo della figura vada il raggio visuale,
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diff changeset
2719 il che non si vede nel conio. </s>
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diff changeset
2720 <s>Questa difficoltà facilmente si toglie avvertendo che ogni volta che miriamo
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diff changeset
2721 alcuna cosa e sia di qualunque figura non vediamo solamente l’obbietto, ma
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diff changeset
2722 ancho assai maggiore spatio intorno a esso, il quale è come un cerchio che è
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diff changeset
2723 base del conio la cui ponta è nel centro dell’occhio, come insegna Eliodoro
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diff changeset
2724 Larisseo nella Prospettiva. </s>
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diff changeset
2725 <s>Il qual conio è tutto ripieno di <pb pagenum="folios 49v-50r"></pb>luce, onde
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diff changeset
2726 forse per questa ragione Averroe nel lib. de sen. et sen. disse esser propio
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diff changeset
2727 del vedere haver la presenza del cono lucido, cioè dell’obbietto illuminato
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2728 con questo conio luminoso. </s>
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2729 <s>La qual cosa non intendendo bene alcuni Filosofi antichi, secondo che
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2730 racconta Aristotile nel lib. </s>
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diff changeset
2731 <s>De sen. et sen., cap. 2, dissero ‘l vedere esser fuoco. </s>
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2732 <s>Il che appresso di esso nel med. lib. nel cap. 2 dimostra Empedocle, facendo
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diff changeset
2733 comparatione del fuoco del vedere, cioè della luce degli occhij al lume
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2734 racchiuso nella lanterna, come si vede ne’ suo’ versi.</s>
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2735 </p>
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2736 <p type="main">
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2737 <s>Seu casu quis progredi meditans lanternam preparavit</s>
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2738 </p>
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2739 <p type="main">
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2740 <s>Hybernam per noctem ignis ardentis lumen </s>
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2741 </p>
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2742 <p type="main">
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2743 <s>accendens splendidum cornu omnimodi obstaculum flatus,</s>
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2744 </p>
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2745 <p type="main">
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2746 <s>Quod ventorum quidem flantium dissipat spiritum.</s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2747 </p>
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2748 <p type="main">
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2749 <s>Extrinsecus autem extensum exiliens quantum procurrevit lumen,</s>
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2750 </p>
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2751 <p type="main">
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2752 <s>Splendevit per pavimentum indomitis radijs,</s>
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2753 </p>
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2754 <p type="main">
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2755 <s>Sic quod in tuniculis costringitur antiquis ignis.</s>
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2756 </p>
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2757 <p type="main">
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2758 <s>Subtilibus velis rotundam complectitur pupillam</s>
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2759 </p>
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2760 <p type="main">
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2761 <s>Que circumfluentis profundum continet aque.</s>
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2762 </p>
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2763 <p type="main">
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2764 <s>Ove si vede espressamente che ‘l fuoco o lo splendore compreso nella pupilla
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2765 dell’occhio è quasi un lume racchiuso nella lanterna essendo contenuto dalle
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2766 torricelle dell’occhio: e come la luce chiusa nella lanterna penetra l’osso,
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2767 il talco, o ‘l vetro e illumina il luogo oscurato dalle tenebre della notte
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2768 l’Inverno; così la <lb></lb>//<lb></lb>detta luce natia dell’occhio trapassa oltre
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2769 agli altri humori, ancho l’humor aque; perciochè ancho ‘l Filosofo nel med.
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2770 luogo disse la pupilla e l’occhio esser acque per cagion degli humori che
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2771 concorreno a formarlo; ma riguardando alla luce contenuta in esso che da
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2772 Empedocle e dagli antichi è appellata fuoco, disse pur nello stesso luogo la
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2773 pupilla essere come il lume della lanterna, la quale rotta si fanno le
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2774 tenebre. </s>
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2775 <s>Onde possiamo ritrarre che quella cosa luminosa che si contiene nel conio
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2776 della vista sia ‘l lume della pupilla. </s>
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2777 <s>Quindi adunque si ritrahe lo scioglimento del dubbio che se s’apre il conio
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2778 de’ raggi visuali ha per base il cerchio, benchè spesse volte la figura
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2779 veduta sia angolare perciochè lo spazio che circonda l’obbietto è sempre di
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2780 questa figura. </s>
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2781 <s>Ma per tornare al proposito nostro diciamo pure che l’uso dell’angolo nella
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parents:
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2782 prosp. si conosce per l’uso del conio, che è effetto dell’angolo, perciochè
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parents:
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2783 dall’angolo prende origine. </s>
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parents:
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2784 <s>E che ciò sia vero si rimiri, che tutto quel che si vede dagli occhi nostri
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parents:
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2785 si vede per virtù del conio fatto da’ raggi visuali; onde, perché talvolta
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parents:
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2786 ha l’angolo minore, talvolta maggiore e tal’hora ancho uguale, perciò è
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parents:
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2787 cagione che l’obbietto ci apparisca tal’hora uguale, tal’hora maggiore e
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parents:
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2788 talvolta minore. </s>
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parents:
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2789 <s>Onde Euclide nella .5a. supposizione disse: Quelle cose che sotto maggior
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2790 angolo si veggono ci appariscono maggiori. </s>
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2791 <s>E nella .6a.: Quelle che sotto minor angolo si veggano appariscono minori. </s>
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parents:
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2792 <s>E nella .7.: Quelle che sotto uguale angolo si veggano appariscono uguali. </s>
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2793 <s>Di modo che quindi si cono<pb pagenum="folios 50v-51r"></pb>sce che tutto quel
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2794 che si vede apparire sotto qualche angolo; che la figura formata da’ raggi
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2795 visuali e dall’obbietto per cagionar la visione, non è altro che ‘l conio
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parents:
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2796 sicome s’è detto. </s>
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2797 <s>Il quale ci mostra l’obbietto maggiore o minore, secondo la grandezza
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2798 dell’angolo da esso costituito, perciochè la grandezza degli angoli e la
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2799 proportione che è fra un angolo e l’altro ritratta dalle portioni delle
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2800 circonferenze del conio è cagione che l’obbietto si mostri hora minore ed
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2801 hore maggiore. </s>
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2802 <s>E sicome del crescimento dell’angolo si cagiona il crescimento dell’apparenza
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2803 della grandezza dell’obbietto, così all’incontro, dallo scemamento, si
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2804 cagiona lo scemamento della grandezza; perciochè la medesima forza che hanno
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2805 gli angoli nella Geometria, di aggiongere o scemar gli spatij (sì come altre
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2806 volte s’è detto) ritengono anchora nella Prospettiva; ma si adatta a diverso
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diff changeset
2807 fine perciochè quivi si applica solamente per ritruovar l’uguaglianza o la
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parents:
diff changeset
2808 disuguaglianza delle grandezze pure geometriche e qui si adatta solamente
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2809 alla diversa apparenza degli obbietti visibili. </s>
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parents:
diff changeset
2810 <s>Oltre acciò, quando l’angolo è maggiore o minore, comprende più o meno dello
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2811 spatio dello splendor del conio posto intorno all’obbietto; di maniera che
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2812 quando l’angolo sarà uguale le cose saranno vedute con ugual lume; onde
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2813 necessariamente ci appariranno uguali. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2814 <s>Talchè quindi anchora si ritrahe che la grandezza dell’angolo porge la
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2815 grandezza dell’apparenze dell’obbietti. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2816 <s>Il medesimo effetto si può vedere, collocato l’<lb></lb>//<lb></lb>obbietto in varie
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2817 positure, come si chiarisce da Euclide nell’ottava supposizione, là dove si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2818 vede che quelle cose che poste in alto fanno conio con maggior angolo si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2819 mostrano più basse, cosa che si rende certa per l’esperienza del Sole, che
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2820 nell’inverno apparisce più basso. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2821 <s>Di più, non solamente l’angolo del conio con la varia sua grandezza è cagione
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2822 d’apparenze diverse, ma ancho quanto più è diviso o moltiplicato in più
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2823 angoli secondo che un conio maggiore si divide in più conij minori, tanto
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2824 più è cagione di più perfetta visione. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2825 <s>E però nella dodicesima supp. disse: quelle cose che si vedono sotto più
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2826 angoli si vedono più distintamente. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2827 <s>E la ragione si è che ciò che si vede non si può veder se non sotto qualche
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2828 angolo, come è già manifesto, allhora si vedrà più perfettamente, quando
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2829 sarà contenuto sotto più angoli, perciò che essendo più angoli saranno più
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2830 conij ed essendo più conij saranno ancho molti più raggi visuali, onde se i
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2831 primi raggi visuali arrivano alla estremità dell’obbietto, i secondi, i
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parents:
diff changeset
2832 terzi, i quarti, i quinti e così altri anchora feriranno l’obbietto dentro
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2833 l’estremità, di maniera che non vi sarà parte d’obbietto alla quale non
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2834 gionghino i detti raggi. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2835 <s>Onde se quelle cose si veggono alle quali giongonsi i raggi visuali, come
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2836 disse nella terza supp. quelle più esattamente si vedranno alle quali
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2837 giungeranno più raggi visuali; come se tuttta la vista apprensiva del vedere
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2838 proceda muovendosi sopra ‘ raggi visuali; perciochè tutta la virtù del
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2839 vedere secondo Eliodoro è collocata in <pb pagenum="folios 51v-52r"></pb>quelle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2840 cose che li son davanti, le quali sono i raggi visuali, il cono e l’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2841 obbietto in quanto è base del conio conforme a questa sentenza è ‘l detto
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2842 del Filosofo nel 2° delle parti degli animali, cioè che ‘l vedere è stato
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2843 collocato davanti perciochè si discerne dirittamente el movimento del vedere
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2844 si fa davanti e nel medesimo modo si vede che si fa ‘l movimento. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2845 <s>Ma per provar l’uso degli angoli nelle dimostrazioni de’ diversi effetti
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2846 dell’apparenze proporremo, se non tutti, almeno alquanti accidenti, che
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2847 avvengono alle varie apparenze. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2848 <s>Con ciò sono l’obbietto non potersi veder tutto in una volta, il che si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2849 dimostra da Euclide nel 1° teorema della Prosp. e la ragione di questo
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2850 accidente si è lo intervallo infraposto a’ raggi visuali; onde avviene essi
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2851 non ferischino l’obbietto continuamente. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2852 <s>Né ciò può accadere senza l’aiuto dell’angolo, perciochè se i raggi visuali
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2853 non formassero angolo nel mezzo dell’occhio non farebbero ‘l conio e non
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parents:
diff changeset
2854 facendolo non riceverebbero la distanza in fra loro, per la quale l’obbietto
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2855 non si vedesse tutto in un girar d’occhio. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2856 <s>Se adunque l’angolo è cagion dell’intervallo de’ raggi visuali e l’intervallo
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2857 è cagione che l’obbietto non sia veduto tutto in una volta; ma prima una
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2858 parte e poi l’altra come accade nel rimirar leggendo qualche scrittura,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2859 l’angolo anchora ne sarà cagione e tanto più, quanto più sarà maggiore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2860 perciochè sarà cagion di maggior discostamento de’ raggi visuali. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2861 <s>E però, acciò più esattamente <lb></lb>//<lb></lb>si veda alcuna cosa, si divide
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2862 tutto lo intero conio in più conij minori come s’insegna nella dodicesima
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2863 supp. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2864 <s>Onde poi tante volte si vede un obbietto in quanti conij si divide il conio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2865 maggiore ed in quante parti lo stesso obbietto è diviso; perciochè tante
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2866 basi son quanti conij; Di modo che la prima vista forma un conio che riceve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2867 la base in una parte dell’obbietto: la seconda ne forma un altro che si fa
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2868 base d’un’altra parte dell’obbietto e così seguendo finchè si scorra co’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2869 raggi visuali per tutta la sua grandezza: o vero diciamo che ‘l medesimo
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2870 conio venga replicato e mosso sopra a ciascuna parte dell’obbietto
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2871 successivamente. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2872 <s>Onde segue che tutto l’obbietto in un solo aprir d’occhio non si possa veder;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2873 ma faccia bisognio o moltiplicar i conij e così ancho gli angoli o muover
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2874 sopra l’obbietto il medesimo conio più volte anzi insieme col conio replicar
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2875 il raggio visuale che è asse di esso e ferir continuamente nel mezzo della
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2876 base del conio, cioè nel mezzo della parte dell’obbietto che è veduta a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2877 guisa di coloro che tirano di mira che non feriscono ‘l bersaglio se non
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2878 trovano ‘l suo centro. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2879 <s>Le grandezze di qualsivoglia obbietto benchè uguali non apparire uguali. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2880 <s>Accidente che si mostra da Euclide nel 5° teorema. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2881 <s>Supposto che disugualmente sieno lontane dall’occhio e tutta la ragione di
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2882 esso consiste nella grandezza degli angoli sotto ‘quali son vedute e colla
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2883 conferenza degli angoli si dimostra. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2884 <s>Il medesimo anchora si dimostra nel 7° teorema. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2885 <s>Gli angoli delle grandezze uguali, poste disugualmente lontane all’occhio non
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2886 tengono la medesima proportione che è nelle distanze. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2887 <s>Il che si vede nel teorema 8°. Perciochè è minore la proportione del maggior
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2888 <pb pagenum="folios 52v-53r"></pb>angolo sotto cui è veduta la grandezza più
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parents:
diff changeset
2889 vicina; al minore sotto ‘l quale è veduta la grandezza più lontana, che non
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2890 è l’intervallo maggiore della grandezza lontana. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2891 <s>Non potersi veder alcuna ragion d’angoli posti lontani alla vista. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2892 <s>Quest’accidente si manifesta nel teorema 9°, là dove si legge: Le grandezze
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2893 rettangole che di lontano son vedute appariscono ritonde. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2894 <s>La cagione di quest’effetto si cerca da Alessandro Afrodiseo nel 37° Pub. </s>
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2895 <s>Ma se ne rende la ragione più filosofica che matematica ed è che l’occhio non
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diff changeset
2896 può da lontano veder gli angoli, essendo sotto li iguali levati, ciò che
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diff changeset
2897 rimane apparisce ritondo. </s>
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2898 <s>Ma secondo la prospettiva avviene altramente; perciochè questo effetto
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2899 avviene sì per cagion della distanza che è fra l’obbietto e l’occhio, sì
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diff changeset
2900 anchora per cagion della figura angolare. </s>
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diff changeset
2901 <s>Dalla figura angolare nasce ‘l non si poter vedere gli angoli di lontano,
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diff changeset
2902 perciochè sempre la larghezza della figura è minore appresso gli angoli che
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diff changeset
2903 altrove; onde non è maraviglia se di lontano gli angoli non apparischino e
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2904 così ancho le parti vicine agli angoli; e quindi è che di due linee se ne
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diff changeset
2905 forma una sola. </s>
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diff changeset
2906 <s>Dalla distanza procede lo svanimento degli angoli perciochè ogni cosa
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diff changeset
2907 visibile ha una determinata distanza la quale passata non si può più vedere
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2908 e ciò facilmente avviene agli angoli che sono di pochissimo estendimento. </s>
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diff changeset
2909 <s>Perciochè fra le conditioni che si richiedono a formar la visione una si è la
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diff changeset
2910 debita lontananza che è fra l’obbietto e l’occhio e però ‘l Filosofo nel
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diff changeset
2911 cap. 8° De sen. et sen. disse: Il vedere <lb></lb>//<lb></lb>cagionarsi in una
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2912 distanza tale che in essa sia collocata qualche cosa che sia prima e qualche
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2913 cosa che sia ultima oltre alla quale non si possa discernere che non è altro
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2914 che dire che alla perfetta visione si richieda la determinata distanza
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2915 dell’occhio all’obbietto. </s>
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2916 <s>E per questa ragione Alessandro Afrodiseo nel 4° della Metafisica, nel comm.
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diff changeset
2917 .55. c.115, diceva che allhora il vedere dimostra la verità quando si rimira
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2918 da un intervallo mediocre non quando da lontano. </s>
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2919 <s>Ma torniamo a considerar gli accidenti dell’apparenze. </s>
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2920 <s>L’occhio che dirittamente alla linea retta, od a cosa che per linea retta si
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diff changeset
2921 erga sopra ‘l piano non poter veder la sua longhezza. </s>
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2922 <s>La qual cosa si afferma da Egnatio Danti con l’autorità di Vitellione
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2923 nell’annotatione del 22° teorema della Prosp. </s>
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2924 <s>Ma la ragione di questo si è perciochè l’occhio vede la linea in un punto,
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2925 onde concorrendo i raggi visuali in esso e confondendosi insieme, non posson
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2926 formare il conio e per conseguenza neancho l’angolo; perciochè ‘l punto non
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2927 è atto ad esser base del conio, essendo indivisibile; e se pure è
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2928 divisibile, essendo punto di Prosp., cioè visuale, in ogni modo non può
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2929 essere bastevole ad esser base del conio, onde segua un angolo che sia buono
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2930 a far discerner l’obbietto essendo d’un’acutissima e strettissima grandezza. </s>
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2931 <s>E se per avventura l’occhio starà a piombo sopra una colonna, non potrà veder
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2932 la sua longhezza, poiché la vedrà tutta nella superficie più vicina, la
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2933 quale sarà al base conio. </s>
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2934 <s>Il medesimo effetto si potrebbe sperimentare collocando l’occhio e la linea e
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2935 la colonna nel me<pb pagenum="folios 53v-54r"></pb>desimo piano, in maniera che
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2936 stieno per diritto fra loro. </s>
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2937 <s>Oltre acciò talvolta essendo opposta una palla all’occhio nostro, vedremo di
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2938 lei la parte minore parerci maggiore e per opposito la parte maggiore
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2939 dimostrarsi minore e ‘l tutto accade per la forza dell’angolo del conio che
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2940 si fa maggiore o minore discostandosi od accostandosi all’obbietto, come si
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2941 dimostra nel teor. 24°. L’istesso accade alla colonna, come si dimostra nel
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2942 30°. Così ancho avviene al conio alzato o abbassato l’occhio, come apparisce
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2943 nella dimostratione del 34°. E tutti questi casi son cagionati dalla
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2944 grandezza dell’angolo, sì chome ancho l’ugualità degli angoli formati da’
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2945 raggi visuali nel mirar i diametri d’un cerchio il cui centro stia a piombo
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2946 sotto l’occhio è cagione che si vedano i detti diametri uguali, sì come è
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2947 manifesto per la dimostratione del 36° teor. e tutta la ragione di questo
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2948 effetto consiste nella linea perpendicolare dall’occhio sopra ‘l piano del
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2949 cerchio la qual faccia angoli retti per ogni verso; onde segue che l’occhio
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2950 col suo centro ferisca ‘l centro del cerchio e perciò discerna tutto
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2951 ugualmente; poiché ciascuna parte di esso li è ugualmente lontana. </s>
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2952 <s>La qual cosa non succederebbe se l’occhio non fusse collocato sopra la linea
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2953 perpendicolare perciochè gli angoli del conio sarebbero disuguali sì come
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2954 anchora quegli del piano del cerchio e gli angoli disuguali (come si mostra
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2955 nel Teor. 37°) son cagione delle disuguali apparenze dei deti diametri. </s>
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2956 <s>Quindi adunque è manifesto che le diversità dell’ apparenze si dimostrano con
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2957 l’aiuto <lb></lb>//<lb></lb>degli angoli sicome ancho della virtù degli angoli si
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2958 cagionano. </s>
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2959 <s>Perciochè ovunque si fa dimostration di Prosp. sempre si presuppongono
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2960 l’occhio e ‘ raggi visuali terminanti nella grandezza veduta. </s>
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2961 <s>Onde necessariamente risulta l’angolo e ‘l conio di grandezza minore o
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2962 maggiore o uguale secondo la varia positura dell’obbietto e dell’occhio.</s>
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2963 </p>
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2964 <p type="main">
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2965 <s></s>
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2966 </p>
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2967 </chap>
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2968 <chap>
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2969 <p type="head">
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2970 <s>Se ‘l colore è propio obbietto del vedere ond’è che Euclide mostra la
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2971 grandezza esser obbietto?</s>
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2972 </p>
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2973 <p type="head">
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2974 <s>Cap. 14</s>
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2975 </p>
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2976 <p type="main">
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2977 <s>Par cosa non convenevole, affermar nella Prospettiva, che l’obbietto propio
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2978 del vedere sia la grandezza e non il colore; perciochè ella suole adattar
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2979 alle cose da lei considerate, l’esser visibili, onde avviene che si
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2980 distingua dall’altre parti della Matematica. </s>
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2981 <s>La qual cosa essendo avvertita dall’arcivescovo di Cantauria nella Prosp. </s>
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2982 <s>Commune, nella 2a conclusione, disse che ‘l colore illuminato opera
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2983 impressivamente nel vedere, stimando il colore esser obbietto propio di
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2984 esso. </s>
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2985 <s>E sse gli obbietti di qualunque specie sieno si fanno visibili col mezzo del
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2986 colore, per qual cagione ancho il colore non è obbietto propio del vedere?
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2987 Ansi più propio sarà il colore che primiero le si afferisce; ma la
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2988 grandezza, se però è propio obbietto del vedere, dopo ‘l colore si apprende. </s>
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2989 <s>Oltre acciò, se Arist. nella part..63. del 2° dell’Anima afferma l’obbietto
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2990 propio d’un senso esser quello che non si apprende da altro senso, la
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2991 grandezza non potrà dirsi obbietto propio del vedere; perciochè com’egli
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2992 stesso nella part. .64. dice è obbietto sensibile commune. </s>
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2993 <s>Oltre acciò, se la grandezza <pb pagenum="folios 54v-55r"></pb>non si apprende
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2994 dagli occhi nostri se non con l’aiuto del colore di cui è vestita, segue che
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2995 non possa esser obbietto sensibile per se stesso e perciò non sarà
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2996 propiamente obbietto sensibile del vedere, come si può confermare con le
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2997 parole della particella .65.. Ma a così fatto quesito potremo per avventura
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2998 rispondere in più modi. </s>
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2999 <s>Perciochè ‘l secondo Filosofo nel lib. </s>
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3000 <s>De sen. e sen., nel 3° capitolo. </s>
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3001 <s>Il colore o è nella stremità del corpo overo è la stessa estremità sua. </s>
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3002 <s>Onde i Pittagorici appellavano la superficie colore. </s>
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3003 <s>Di modo che, in questa maniera il colore essendo la stessa estremità, cioè la
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3004 superficie del corpo, sarà ‘l medesimo che la grandezza; onde Euclide,
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3005 ponendo la grandezza per obbietto del vedere, non escluderà ‘l colore; che
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3006 ‘l colore (come ‘l Filosofo stesso afferma nel medesimo luogo) è l’estremità
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3007 della cosa luminosa in un corpo terminato. </s>
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3008 <s>Oltre acciò si dee considerare che ‘l colore e la grandezza appariscono una
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3009 stessa cosa, perciochè si vedano insieme; che ‘l colore non si dicendone
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3010 separatamente, né la grandezza è invisibile, come sarebbe se non havesse ‘l
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3011 colore; onde se amendue si mostrano una stessa cosa, per questa ragione non
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3012 sidee credere che dicendo Euclide, la grandezza è l’obbietto del vedere, non
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3013 includa ancho il colore; che qualunque faccia mesione solamente della
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3014 grandezza, virtualmente intende ancho il colore, ponendo la grandezza per
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3015 obbietto principale in quanto appartiene alla prospettiva e ‘l colore come
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3016 obbietto congionto, cioè quello stimando suo principale obbietto che più li
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3017 è proportionato. </s>
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3018 <s>Finalmente Euclide considera solamente la grandezza come <lb></lb>//<lb></lb>visibile
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3019 e non riguarda ‘l colore come obbietto; ma come mezzo, pel quale ella si
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3020 apprende; perciochè non si cura della qualità, che non è propio obbietto
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3021 della Prospettiva, ma della quantità e si serve della qualità, dico del
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3022 colore, come di strumento; perciochè Aristotele, nel .p°. cap. de sen. et
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3023 sen., dice tutte le cose corporee esser colorate; di maniera che gli
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3024 obbietti communi col mezzo del colore si conoscano, in fra ‘ quali
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3025 principalmente sono le figure e le grandezze e in questo modo mi par che
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3026 abastanza si sia risposto alla questione proposta.</s>
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3027 </p>
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3028 <p type="main">
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3029 <s></s>
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3030 </p>
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3031 </chap>
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3032 <chap>
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3033 <p type="head">
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3034 <s>L’uso degli angoli nella prospettiva appartenente agli specchij e a’
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3035 reflettimenti de’ raggi del sole</s>
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3036 </p>
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3037 <p type="head">
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3038 <s>Cap. 15</s>
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3039 </p>
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3040 <p type="main">
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3041 <s>Sì come la stessa Prospettiva servendosi delle cose geometriche e
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3042 specialmente della linea e del ponto, dal modo di considerare prende ‘l nome
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3043 col quale determina e specifica (per così dire) le dette cose a se stessa
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3044 applicate, così mentre si serve degli angoli, dalla maniera di considerare e
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3045 dall’adattamento di essi a’ suoi propri termini si trahe ‘l nome; di modo
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3046 che, sì come nomina le linee e punti visuali, così appella gli angoli non
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3047 solamente visuali, ma dell’Incidenza e del Reflettimento. </s>
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3048 <s>Di modo che, come vedo nella Prosp. le cose della geometria con la gionta di
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3049 qualche nome, o conditione, si comprende l’uso di esse: così dall’Incidenza
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3050 e dal Reflettimento adattato agli angoli si prende la cognition dell’uso
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3051 loro nella Prosp. </s>
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3052 <s>Perciochè ciascun arte o scienza che si serve de’ termini, o della materia
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3053 d’altra scienza od arte superiore appropiandogli a se stessa, gli veste di
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3054 nuovi nomi e di nuove conditioni, come si può<pb pagenum="folios 55v-56r"></pb>vedere esaminando tutte le scienze ed arti. </s>
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3055 <s>Gli angoli adunque che si adoperano in questa parte di Prosp.son que’ che si
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3056 appellano dell’Incidenza e del Reflettimento, i quali si formano da’ raggi
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3057 del Sole altresì negli specchi, nell’acqua ed in altra maniera di corpi ne’
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3058 quali s’incontrano. </s>
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3059 <s>Di questi ragionaremo in questo luogo, ma avanti che cominciamo, per
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3060 continuar in parte il presente ragionamento, con le cose dette, bisogna
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3061 avvertire che la parte della Prosp. ottica ragiona de’ raggi visuali in
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3062 quanto appartengono al vedere gli obbietti dirittamente nel mezzo
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3063 illuminato; che pertanto Euclide nella sua Prosp. fece questo supposto, che
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3064 ‘raggi che escon dall’occhio si muovono per retta linea, como ancho afferma
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3065 Eliodoro Larisseo e l’Arcivescovo di Cantauria nella concl. 48 della prima
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3066 parte della Prop. </s>
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3067 <s>Comm., ove dice, facilmente vedersi gli obbietti dirittamente. </s>
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3068 <s>Ma nel mezzo illuminato; perciochè questo stesso autore nella concl. 49
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3069 afferma che nissuna cosa si può vedere senza lume. </s>
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3070 <s>Il che oltre alla sperienza, si conferma dal filosofo nel 2° dell’Anima nella
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3071 part. 67 là dove afferma che ‘l colore non si può veder senza ‘l lume el
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3072 mezzo nel quale si vede non essendo illuminato non può render colore alcuno. </s>
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3073 <s>Oltre che secondo Francesco Piccolomini nel Duodo nel cap. 5 del 6. Il lume
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3074 si trova negli obbietti per cagion del mezzo, pel quale passa; che non può
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3075 giogner da uno estremo, cioè dal corpo luminoso all’altro estremo, cioè
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3076 all’obbietto senza passar pel mezzo. </s>
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diff changeset
3077 <s>La parte specularia è tutta volta alla consideratione e all’uso
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diff changeset
3078 <lb></lb>//<lb></lb>de’ raggi del lume e de’ visuali. </s>
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parents:
diff changeset
3079 <s>Perciochè se l’ottica considera i raggi visuali in quanto si adoperano alla
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parents:
diff changeset
3080 vista dirittamente fatta, poiché procede secondo la dirittura de’ raggi che
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parents:
diff changeset
3081 uscendo dall’occhio e terminando nell’obbietto formano ‘l conio; la
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parents:
diff changeset
3082 specularia all’incontro ragiona de’ raggi visuali in quanto appartengono al
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parents:
diff changeset
3083 vedere per modo reflesso negli specchi illuminati insieme col mezzo. </s>
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parents:
diff changeset
3084 <s>Per modo reflesso dico, perciochè la vista in essi vien regolata secondo ‘l
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parents:
diff changeset
3085 reflettimento de’ raggi che si partano dall’occhio per andar a trovar
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parents:
diff changeset
3086 l’obbietto e la sua immagine impressa negli specchi, ne’ quali, essendo
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parents:
diff changeset
3087 piani o ritondi e convessi e concavi (secondo Eucl. nel primo teor.) i raggi
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diff changeset
3088 visuali si ripiegano ad angoli uguali, come s’avvertisce da Ignatio Danti
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parents:
diff changeset
3089 sopra la terza supp. d’Eucl. negli specchi e per la gran conformità posta in
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diff changeset
3090 fra la luce e gli occhi (onde è che da pochi sono stati chiamati lumi,
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parents:
diff changeset
3091 splendori, stelle e soli) ciò che de’ raggi visuali s’è detto dirsi dee de’
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3092 raggi della luce del Sole, anzi principalmente di questi. </s>
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diff changeset
3093 <s>Perciochè ancho essi non solamente negli specchi qualunque sieno; ma anchora
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parents:
diff changeset
3094 ne’ corpi lucidi, come nelle gioie e nelle pietre egregiamente polite anzi
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3095 nella terra anchora, benchè opaca e varia di colori e di superficie; si
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parents:
diff changeset
3096 reflettano con varie specie d’angoli ed in ogni ripiegamento formano
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diff changeset
3097 dall’una e l’altra banda angoli uguali, poiché sempre si ripiegano in un
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3098 punto. </s>
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parents:
diff changeset
3099 <s>Ma bisogna avvertire che benchè i raggi visuali e ‘ raggi del Sole
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parents:
diff changeset
3100 riflettendosi negli specchi formino angoli uguali, con tutto ciò in fra essi
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parents:
diff changeset
3101 è qualche differenza, che altro è l’angolo della Reflession de’ raggi
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parents:
diff changeset
3102 visuali <pb pagenum="folios 56v-57r"></pb>altro de’ raggi del Sole negli
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parents:
diff changeset
3103 specchi, e la differenza si prende dal fine , il quale in essi è diverso;
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parents:
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3104 perciochè l’angolo de’ raggi visuali appartiene al veder le cose negli
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diff changeset
3105 specchi; ma l’angolo de’ raggi del Sole appartiene al raddoppiamento degli
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diff changeset
3106 splendori de’ lumi e della vista che è atta ad aveder le materie oppostegli. </s>
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3107 <s>Ma per cominciar a scoprir l’uso degli angoli nella Specularia, seguendo di
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parents:
diff changeset
3108 farne comparatione all’ottica, diremo che se nell’ottica si ritrahe l’uso
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3109 degli angoli in quanto son fatti da’ coni de’ raggi visuali procedenti
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parents:
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3110 dirittamente pel mezzo dell’obbietto, acciò si formi la visione: nella
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parents:
diff changeset
3111 specularia si conosce l’uso di essi come formati da’ raggi visuali, che non
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parents:
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3112 vanno dirittamente all’obbietto, overo alla immagine sua; ma per modo
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3113 ripiegato; perciochè vanno dirittamente fine alla superficie dello specchio,
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3114 là dove formano gli angoli dell’Incidenza; e dallo specchio si riperquotano
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3115 e fanno gli angoli reflessi, acciochè i raggi del conio della veduta
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diff changeset
3116 prendano l’immagine dell’obbietto che è fuor dello specchio, acciò la vista
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parents:
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3117 habbia l’uso del suo conio nell’obbietto reale e nell’immagine, talmente che
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3118 la base che è nell’immagine corrisponda alla base che è nell’obbietto;
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3119 perciochè ripiegandosi i raggi visuali nello specchio e nella simiglianza
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3120 stessa dell’obbietto finalmente si terminino nell’obbietto, onde prende
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3121 origine tutto quel, che apparisce nello specchio. </s>
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3122 <s>La qual cosa vedrà apertissimamente chiunque osservarà Eucl. nella Spec. e
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diff changeset
3123 specialmente nelle dimostrationi de’ teoremi .4.7.11.17. e .20. Di modo che
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parents:
diff changeset
3124 nell’ottica si scorgano solamente gli angoli dell’Incidenza, che son quelli
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parents:
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3125 che si <lb></lb>//<lb></lb>formano da’ raggi visuali mentre perquotono
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parents:
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3126 nell’obbietto, costituendolo base del conio loro. </s>
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3127 <s>E se favellando alla peripatetica, dicessemo i raggi visuali nascer
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3128 dall’obbietto , costituiremo un angolo solo dell’Incidenza formato nel
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3129 centro dell’occhio. </s>
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3130 <s>Però si dice che la vista procede per linea retta. </s>
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3131 <s>Ma nella Prosp. degli specchi oltre agli angoli dell’Incidenza si veggono
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parents:
diff changeset
3132 quegli del reflettimento. </s>
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parents:
diff changeset
3133 <s>Perciochè ogni volta che i raggi visuali si riperquotano negli specchi fanno
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parents:
diff changeset
3134 due angoli, l’uno dell’Incidenza, che è ‘l primo che si fa nel ferir lo
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parents:
diff changeset
3135 specchio e l’altro della reflessione, che è ‘l secondo fatto tosto doppo ‘l
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parents:
diff changeset
3136 primo sorgendo e muovendosi finchè gionghino al termine loro, che è
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parents:
diff changeset
3137 l’obbietto reale. </s>
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parents:
diff changeset
3138 <s>L’angolo dell’Incidenza è quello che si comprende dalla superficie dello
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3139 specchio e della linea che esce dall’occhio. </s>
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parents:
diff changeset
3140 <s>L’angolo della Reflessione è quello che si abbraccia dalla superficie dello
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parents:
diff changeset
3141 specchio e dalla linea che si ripiega dal punto dell’incidenza che è quello
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parents:
diff changeset
3142 in cui ferisce il raggio visuale e si muove verso altro punto in cui si
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parents:
diff changeset
3143 ferma. </s>
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parents:
diff changeset
3144 <s>Di maniera che amendue questi angoli si formano sopra la superficie dello
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parents:
diff changeset
3145 specchio e si suppongano uguali acciò possano servire alle dimostrationi ,
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parents:
diff changeset
3146 come si può vedere, osservando Eucl. </s>
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parents:
diff changeset
3147 <s>Oltre a questi angoli vi è ancho ‘l terzo esterno e maggiore d’amendue, il
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parents:
diff changeset
3148 quale è contenuto dal raggio diritto e dal raggio piegato. </s>
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parents:
diff changeset
3149 <s>Senza i quali angoli non si può veder cosa alcuna negli specchi; perciochè,
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parents:
diff changeset
3150 come dice Eucl. nel Teor. 18° e 27°, le immagini delle cose visibili negli
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parents:
diff changeset
3151 specchi, si vedano in quel luogo ove concorre il raggio che esce dall&#039;’cchio
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parents:
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3152 con la linea, che dalla cosa visibile si è tirata fine al centro dello
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parents:
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3153 specchio <pb pagenum="folios 57v-58r"></pb>concavo; che sì come afferma Ignatio
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parents:
diff changeset
3154 Danti sopra la quarta supp. e apparenza prima, le cose che si vedono negli
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parents:
diff changeset
3155 specchi non si vedano per quella linea con cui s’improntano, ma nel concorso
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parents:
diff changeset
3156 di detta linea e del raggio che esce dall’occhio e ciò avviene quando il
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parents:
diff changeset
3157 raggio non si ripiega in se stesso come si mostra da Eucl. nel 2° Teor.
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parents:
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3158 perciochè in questo caso la linea dell’incidenza e della reflessione è
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parents:
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3159 l’istessa, onde non vi si scorge se non l’angolo dell’incidenza. </s>
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parents:
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3160 <s>Questo concorso di raggi visuali con le linee degli obbietti non si fa senza
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parents:
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3161 i detti angoli ed in esso si vedano le immagini degli obbietti visibili e
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parents:
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3162 però negli angoli dell’Incidenza si forma la visione di ritratti delle cose
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parents:
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3163 visibili negli specchi; perciochè (come nel precedente capitolo si è
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parents:
diff changeset
3164 dimostrato), ciò che si vede apparisce sempre sotto qualche maniera
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diff changeset
3165 d’angolo; che se nell’ottica ciò si dimostra nel conio diritto, nella Prosp.
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parents:
diff changeset
3166 degli specchi si dimostra nel conio reflesso. </s>
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parents:
diff changeset
3167 <s>Ma si avvertisca che nell’ottica solamente si fa conto dell’angolo costituito
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3168 da’ raggi visuali nel centro dell’humor cristallino e nella specularia oltre
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parents:
diff changeset
3169 a questo si stimano ancho gli angoli formati nella superficie dello specchio
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diff changeset
3170 nella stremità dell’immagine dell’obbietto, là dove si forma la visione
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diff changeset
3171 reflessa. </s>
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parents:
diff changeset
3172 <s>E per intender meglio questo ripiegamento de’ ragi visuali negli specchi fa
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diff changeset
3173 di mestiero considerare che può dirsi che ‘raggi visuali che escono dagli
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parents:
diff changeset
3174 occhi e feriscono lo specchio ne’ termini del diametro dell’immagine
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parents:
diff changeset
3175 impressavi si ripieghino partendosi da’ punti dell’Incidenza per muoversi
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parents:
diff changeset
3176 tanto che gionghino all’obbietto rappresentato dell’imagine dello specchio,
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parents:
diff changeset
3177 acciochè si formi un conio reflesso colla forza del <lb></lb>//<lb></lb>quale si
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parents:
diff changeset
3178 faccia la visione della detta immagine conforme alla vision dell’obbietto
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3179 reale; perciochè la base del conio che è nell’immagine impressa nelo
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parents:
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3180 specchio sarà simile e proportionale alla base che si ritruovarà nello
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3181 stesso obbietto. </s>
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diff changeset
3182 <s>Onde si vede che i raggi visuali che fanno gli angoli dell’incidenza e della
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parents:
diff changeset
3183 reflessione per cagion del conio appariscono gli stessi raggi che escon
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parents:
diff changeset
3184 dagli occhi per formare ‘l conio diritto che la natura loro sarebbe di andar
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diff changeset
3185 dirittamente, ma l’incontro dello secchio gli fa piegare; che tale sia la
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parents:
diff changeset
3186 natura loro, si vede quando s’incontrano in un corpo lucido e trasparente
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3187 come è l’acqua che è quasi uno specchio piano in cui naturalmente i raggi
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parents:
diff changeset
3188 che verso la cosa rappresentata si reflettano per trarne l&#039;’mmagine e
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parents:
diff changeset
3189 trasportarla nell&#039;’cqua benchè a rivescio, si vanno dirittamente continuando
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parents:
diff changeset
3190 tanto che trasportino la figura dell’obbietto nella stessa perpendicolare,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3191 come si vede nelle due dimostrationi del 7° Teor. </s>
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diff changeset
3192 <s>Ma si dee avvertire che’ detti raggi che si continuano nell’acqua non formano
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parents:
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3193 linee perpendicolari, ma più tosto inclinate; che se facessero linee
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3194 perpendicolari si ripiegarebbero in loro stesse e non formarebbero altro
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parents:
diff changeset
3195 angolo che dell&#039;incidenza e però nol farebbero Reflesso; perciochè avviene
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parents:
diff changeset
3196 de’ raggi del vedere non altramente che avvenga di quegli del Sole i quali
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parents:
diff changeset
3197 non si ripiegano sopra ‘l corpo ove perquotono se non quando non son
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3198 perpendicolari, come si vede appresso l’Autor della Prosp. </s>
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parents:
diff changeset
3199 <s>Comm. nella quindicesima concl. della prima parte. </s>
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parents:
diff changeset
3200 <s>Oltre acciò quando Eucl. nel Teor. 18° e 27° dimostra negli specchi all’hora
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parents:
diff changeset
3201 vedersi le immagini delle cose visibili quando concorreno i raggi che escono
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parents:
diff changeset
3202 dagli occhi insieme con le linee, che dalla cosa visibile si tirano; par che
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parents:
diff changeset
3203 accenni che l’obbietto stesso imprime la sua immagine nello spec<pb pagenum="folios 58v-59r"></pb>chio perciochè a qual fine tirar le linee
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parents:
diff changeset
3204 dall’obbietto allo specchio, se l’obbietto non v’imprime la sua figura col
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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3205 mezzo di esse. </s>
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parents:
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3206 <s>Adunque l’obbietto con la forza delle linee che da esso nascono cagiona la
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3207 sua figura nello specchio; Onde pare che in queste Eucl. segua l’oppinion
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diff changeset
3208 de’ Peripatetici e in parte quella de’ Platonici, questa col mostrar che le
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parents:
diff changeset
3209 cose che si discernano nello specchio si vedano per la forza de’ raggi
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parents:
diff changeset
3210 visuali uscenti dagli occhi e quella col dimostrar che le linee nascono
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3211 dall’obbietto. </s>
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parents:
diff changeset
3212 <s>Ma se l’obbietto ha virtù d’imprimer la sua effigie negli specchi, come non
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3213 harà la medesima possanza ancho negli occhi, che sono simigliantissimi agli
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parents:
diff changeset
3214 specchi, sicome si vede per esperienza e come si ritrahe da Francesco
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parents:
diff changeset
3215 Petrarca nella Canz. </s>
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parents:
diff changeset
3216 <s>Perché la vita è breve, là dove si legge:</s>
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parents:
diff changeset
3217 </p>
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parents:
diff changeset
3218 <p type="main">
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parents:
diff changeset
3219 <s>Luci beate e liete,</s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3220 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3221 <p type="main">
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parents:
diff changeset
3222 <s>Se non che ‘l veder voi stesse v’è tolto,</s>
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parents:
diff changeset
3223 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3224 <p type="main">
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parents:
diff changeset
3225 <s>Ma quante volte a me vi rivolgete,</s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3226 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3227 <p type="main">
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parents:
diff changeset
3228 <s>Conoscete in altrui quel che voi sete.</s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3229 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3230 <p type="main">
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parents:
diff changeset
3231 <s>Adunque sì come l’obbietto opera nello specchio, così potrà operare negli
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3232 occhi. </s>
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casties
parents:
diff changeset
3233 <s>Nello specchio principalmente e negli occhi col mezzo dello specchio. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3234 <s>Ma si avvertisca che Eucl. non dice che le linee nascano dall’obbietto, ma
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3235 che si tirano dall’obbietto, acciò si faccia ‘l detto concorso e perciò non
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3236 potrà dirsi, che la veduta si faccia per ricevimento di specie insieme co’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3237 Peripatetici. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3238 <s>Perciochè anchorchè la vista che si fa intorno all’immagini rappresentate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3239 dagli specchi si formi col mezzo del ricevimento delle impronte di esse,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3240 contuttociò non è ‘l ricevimento considerato da Aristotile perciochè non si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3241 fa negli occhi, ma nello specchio, là dove fanno angoli i raggi visuali e si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3242 congiongano con le linee degli obbietti. </s>
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parents:
diff changeset
3243 <s>Perciochè in tal congiognimento si riceve la figura dell’<lb></lb>//<lb></lb>obbietto
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3244 da raggi visuali. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3245 <s>Né per l’uscir che si faria da raggi visuali secondo i Platonici si cagiona
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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3246 la visione negli specchi perciochè non è l’uscir di essi cagion della
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3247 visione, ma ‘l detto concorso de’ raggi visuali con le linee degli obbietti. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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3248 <s>Oltre acciò i raggi visuali per loro stessi giognerebbero all’obbietto se non
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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3249 s’incontrassero nelle linee che si partano da esso per trasportar l’immagine
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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3250 sua nello specchio. </s>
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parents:
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3251 <s>E però l’oppinion d’Eucl. sarà schiettamente Platonica, cioè al tutto lontana
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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3252 dal ricevimento dell’immagine visibile. </s>
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parents:
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3253 <s>Ma per non tralassar quel che appartiene agli specchi dirò che altri son
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3254 piani, altri tondi de’ quali altri sono concavi ed altri convessi. </s>
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parents:
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3255 <s>Gli specchi tondi, o sieno convessi o concavi, sono tutti portioni di cerchi
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diff changeset
3256 considerati nella superficie piana, in cui si figurano; ma considerati
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diff changeset
3257 nell’esser loro non sono altro che tante portioni di corpi sferici, o
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diff changeset
3258 concavi o convessi, non altramente che se noi tagliassemo una palla vota di
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diff changeset
3259 vetro o di rame. </s>
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parents:
diff changeset
3260 <s>Che quantunque Eucl. gli dimostri con portioni di cerchij, come si vede nel
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parents:
diff changeset
3261 Teor. 5.6.8.11.etc. contuttociò si dee intender che le figure da lui formate
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parents:
diff changeset
3262 nel piano della superficie non sien fatte per mostrar simplicemente le
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parents:
diff changeset
3263 portioni de’ cerchij; ma per raffigurar sotto tali portioni i frammenti de’
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diff changeset
3264 corpi sferici. </s>
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parents:
diff changeset
3265 <s>Sono adunque gli specchi portioni di corpi sferici e l’occhio per lo più vien
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diff changeset
3266 collocato dentro la circonferenza loro, o nel centro o infra essa e ‘l
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diff changeset
3267 centro, onde muovendo i raggi visuali verso lo specchio, finalmente con essi
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diff changeset
3268 vi forma l’angolo dell’Incidenza e del Reflesso, mentre vicostituisce ‘l
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diff changeset
3269 conio ripiegato. </s>
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parents:
diff changeset
3270 <s>Gli specchi piani son quegli la cui superficie è in modo <pb pagenum="folios 59v-60r"></pb>giacente e diritta che dall’obbietto la linea
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parents:
diff changeset
3271 vi cade ad angoli retti, sì come vede nelle dimostrationi del Teor. 7.9.16 e
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diff changeset
3272 19 ed essendo allongati fine alla perpendicolare dell’obbietto i raggi
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diff changeset
3273 visuali che già si ripegavano (perciochè secondo il Teor. 16 ciascuna cosa
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parents:
diff changeset
3274 visibile negli specchi piani si vede nella perpendicolare tirata dalla cosa
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3275 visibile fine allo specchio) la superficie dello specchio vien tagliata ad
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diff changeset
3276 angoli disuguali e diversi di specie. </s>
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diff changeset
3277 <s>Onde l’immagine dell’obbietto si trasporta nello specchio al contrario;
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parents:
diff changeset
3278 perciochè essendo già nello specchio impressa l’immagine per mezzo delle
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diff changeset
3279 linee che da esso procedono i raggi visuali che prima si ripiegavano
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3280 prendendo la detta immagine seguono di muoversi finchè quasi penetrando
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diff changeset
3281 l’obbietto trasportino la detta immagine nella perpendicolare, collocandola
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3282 a rivescio in quella guisa che ci appariscono le cose nell’acqua, cioè
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3283 secondo la diversa positurae contraria delle parti; perciochè i raggi
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3284 visuali mutan sito, come si può vedere nel Teor. 7. 19. 11. 12. etc., benchè
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parents:
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3285 sì l’obbietto e sì l’immagine appariscano sotto ‘l medesimo angolo e dentro
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3286 al medesimo conio; che’l conio piegato è equalmente al conio diritto, come
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3287 si può vedere nella dimos. del 7° Teor. </s>
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3288 <s>Ma poiché di sopra si accennò la conformità nella quale convergono gli occhi
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3289 con la luce del Sole e perciò gli splendori della detta luce in diversi
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diff changeset
3290 incontri di corpi se reverberando formar gli angoli della Reflessione e
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3291 dell’Incidenza e quivi cagionar l’ aurescimento del colore e della virtù
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3292 infiammate. </s>
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parents:
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3293 <s>Pertanto fa di mestiero considerare diligentemente tutte le maniere degli
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parents:
diff changeset
3294 angoli che da’ raggi del Sole si producano e la ragion del modo del
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parents:
diff changeset
3295 producimento loro e l’effetto che <lb></lb>//<lb></lb>ne risulta ed insieme anchora
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parents:
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3296 l’uso. </s>
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parents:
diff changeset
3297 <s>Tre sono le maniere degli angoli che si possano produrre da’ raggi del Sole
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parents:
diff changeset
3298 nella ripercussione o nel concorso o nello ‘ncontro loro. </s>
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parents:
diff changeset
3299 <s>Ciò sono gli angoli dell’Incidenza del Reverberamento e dello ‘ntersegamento. </s>
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parents:
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3300 <s>Gli angoli dello Intersegamento son quegli che si formano mentre i raggi del
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parents:
diff changeset
3301 Sole trapassano per uno spiraglio d’un muro o d’una finestra chuisa o d’una
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parents:
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3302 vetrata rotta, perciochè dal corpo luminoso del Sole si partano i raggi i
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parents:
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3303 quali passando pel detto spiraglio s’incontrano insieme e si tagliano
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3304 formando due coni opposti e per conseguenza due angoli opposti uguali. </s>
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3305 <s>Perciochè i raggi che dal Sole si partano, in qualunque punto terminati
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parents:
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3306 secondo l’Autor della Prosp. </s>
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3307 <s>Comm. nella Concl. 5 della prima parte, non possono fermarsi in esso, ma
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parents:
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3308 tagliandosi insieme vanno più oltre; onde quando passano per lo spiraglio
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parents:
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3309 angolare s’intersegano in esso, allongandosi dirittamente e pervenendo da
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parents:
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3310 una parte a tanta lontananza quanta è la lontananza del Sole dall’altra
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parents:
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3311 parte, si allargano secondo la larghezza del Sole perciochè gli angoli
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3312 opposti essendo uguali e ‘ lati altresì uguali, la base anchora sarà uguale,
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parents:
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3313 come si potrebbe confermar per la quarta del primo d’Eucl.: Che ‘ raggi
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parents:
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3314 necessariamente debbano passar più oltre e tagliarsi insieme; quindi è
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parents:
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3315 manifesto perciochè non truovano incontro di corpo denso, che
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parents:
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3316 necessariamente si richiede alla ripercussione de’ raggi, come dimostra ‘l
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3317 medesimo Autore nella concl. .15. della medesima parte: che lo ‘ncontro che
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parents:
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3318 truovano non è altro che un punto del mezzo, cioè dell’aria contenuta nello
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parents:
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3319 spiraglio. </s>
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parents:
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3320 <s>Onde è come punto e come corpo penetrabile da’ raggi del Sole (perciochè
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parents:
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3321 l’aria non sola <pb pagenum="folios 60v-61r"></pb>mente è corpo trasparente, ma
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3322 raro), non è tale che vi si possan fermar o riperquoter i raggi del Sole.Gli
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parents:
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3323 angoli del reverberamento de’ raggi del Sole son simiglianti agli angoli del
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parents:
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3324 Reflesso de’ raggi visuali negli specchi; perciochè son compresi dalla
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parents:
diff changeset
3325 superficie del corpo denso e dalle linee che nascono dal corpo luminoso e
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parents:
diff changeset
3326 incontrandosi nel corpo denso o lucido o chiaro e polito o variamente
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parents:
diff changeset
3327 colorato, com’è la superficie della terra percotendolo nel punto
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3328 dell’Incidenza, quindi si partano e vanno a terminare altrove, cioè in
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parents:
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3329 contraria parte. </s>
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parents:
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3330 <s>E questo avviene perciochè non si può far riflettimento alcuno, così angoli. </s>
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parents:
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3331 <s>E questo accade perciochè non si può far riflettimento alcuno con angoli
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parents:
diff changeset
3332 acuti interni, sì come ancho si vede ne raggi visuali; che la linea
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parents:
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3333 ripiegata non termini in contraria parte obliquamente; perciochè i raggi si
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parents:
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3334 ripiegarebbero in se stessi e cadrebbero nel punto dell’Incidenza
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parents:
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3335 perpendicolarmente e non formarebbero angoli acuti; non retti come si può
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parents:
diff changeset
3336 ritrarre dal 2° Teor. degli specchi. </s>
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3337 <s>E l’autor della Prosp. </s>
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parents:
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3338 <s>Comm. nella quindicesima conclusione della prima parte mostra che ‘l raggio
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parents:
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3339 della luce icontrandosi nel mezzo denso si ripiega, non essendogli
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parents:
diff changeset
3340 perpendicolare. </s>
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parents:
diff changeset
3341 <s>Gli angoli dell’Incidenza de’ raggi del Sole non son diversi da quegli de’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3342 raggi visuali, in quanto al nascimento loro; periochè si producano nel
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parents:
diff changeset
3343 percotimento de’ raggi in qualche corpo denso sì come quegli si cagionano
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parents:
diff changeset
3344 nel ferir lo specchio. </s>
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diff changeset
3345 <s>Onde son que’ che si comprendano dalla superficie del corpo denso e dalla
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diff changeset
3346 linea che nasce dallo stesso corpo luminoso. </s>
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diff changeset
3347 <s>Nel concorso di questi raggi raddoppiata <lb></lb>//<lb></lb>la virtù infiammante si
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diff changeset
3348 accende ‘l fuoco, come si può vedere nella dimostration del Teor. 3 degli
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diff changeset
3349 specchi d’Euclide. </s>
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diff changeset
3350 <s>E ciò basti intorno a questa parte della Prospettiva.</s>
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diff changeset
3351 </p>
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diff changeset
3352 <p type="main">
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diff changeset
3353 <s></s>
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diff changeset
3354 </p>
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diff changeset
3355 </chap>
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diff changeset
3356 <chap>
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diff changeset
3357 <p type="head">
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diff changeset
3358 <s>Dell’uso degli angoli nella Prospettiva scenografica</s>
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diff changeset
3359 </p>
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diff changeset
3360 <p type="head">
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diff changeset
3361 <s>Cap. 16</s>
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diff changeset
3362 </p>
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diff changeset
3363 <p type="main">
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diff changeset
3364 <s>E’ tanto universale l’uso degli angoli che non è parte di Matematica o pura o
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3365 non pura che non si serva di essi. </s>
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diff changeset
3366 <s>Il che si è dimostrato ne’ precedenti cap. come ancho in questo si
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3367 mostrarà.</s>
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diff changeset
3368 </p>
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diff changeset
3369 <p type="main">
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3370 <s> Perciochè nella prospettiva appartenente alle scene l’uso degli angoli non
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3371 solamente è necessario nell’operationi sue; ma ancho nelle ragioni sopra le
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3372 quali tutta la pratica si sostiene. </s>
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diff changeset
3373 <s>E perché ‘l detto uso degli angoli si applica a diverse cose, perciò si
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diff changeset
3374 distingue in varie maniere, dalle quali procedano effetti diversi, come
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diff changeset
3375 nell’operare facilissimamente si conosce.</s>
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3376 </p>
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diff changeset
3377 <p type="main">
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diff changeset
3378 <s>L’uso adunque degli angoli nella scenografia si impiega: nell’accomodamento
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3379 delle cose vere con le finte e nel dar rilievo e dirittura nelle cose poste
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3380 negli angoli concavi e convessi e nel far saltar sopra ‘l piano le cose come
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diff changeset
3381 rilevate: negli intersegamenti varij: nel trasportamento delle figure
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3382 geometriche: e nelle linee del taglio.</s>
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3383 </p>
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3384 <p type="main">
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3385 <s>La linea del taglio che ( come si ritrahe dal primo libro del Cavalier
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diff changeset
3386 Lorenzo Sirigatti nel cap. 3 e 5) altramente è detta delle misure, non si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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3387 può generar senza gli angoli; perciochè è una linea retta perpendicolare,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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3388 che cadendo sopra la linea del piano, la taglia ad angoli retti, sì come è
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3389 natura delle linee perpendicolari: e in un medesimo tempo, tagliando i raggi
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3390 visuali, e le linee che si par<pb pagenum="folios 61v-62r"></pb>tano dal piè
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3391 dell’huomo riguardante, che terminano negli angoli dell’obbietto, dico negli
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3392 angoli della pianta; i visuali ne’ superiori, e le linee pedali negli
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3393 inferiori: e ne’ tagliamenti formano angoli alterni, e opposti uguali, come
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3394 si può vedere appo Euclide nelle propositioni 28 e 29 del primo e ciò si
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3395 vede chiaro; percioché le linee che escono dall’occhio e dal piede, le quali
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3396 toccano gli angoli della figura dell’obbietto, sono fra loro parallele, anzi
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3397 formano due parallelogrammi proportionali, i quali hanno un lato commune
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3398 terminante nel punto dell’occhio e del piede gli angoli de’ quali non
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3399 solamente son simili ma ancho uguali perciochè per la trentacinquesima del
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3400 primo.</s>
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3401 </p>
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3402 <p type="main">
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3403 <s>I parallelogrammi costituiti nella medesima base e nelle medesime parallele
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3404 sono fra loro uguali. </s>
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3405 <s>La linea del taglio è ancho quella che si chiama orizzontale, che (sì come si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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3406 vede appresso Baldassar Peruzzi nel Serlio e appo ’l Vignuova e appresso
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3407 Giovan Cusino) partendosi dal punto principale, in cui si termina la
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3408 lontananza della veduta, e terminando nella linea del piano, il taglia
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3409 proportionalmente e ad angoli uguali, in apparenza ottusi, od acuti ma in
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3410 essenza retti.</s>
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3411 </p>
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3412 <p type="main">
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3413 <s>Può dirsi anchora esser la linea perpendicolare, che si tira nella figura
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3414 geometrica, la quale terminando nella linea del piano, che è base del
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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3415 triangolo, dentro ‘l quale si dispone, e comparte il piano iscorciato, la
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3416 taglia ad angoli retti, e da’ tagliamenti nascendo le linee si congiongano
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3417 in un ponto e intersegano le diagonali con proportion di tagliamenti,
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3418 formando insieme angoli uguali, interni ed esterni e cos
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3419 <lb></lb>//<lb></lb>tituendo col taglio loro il producimento delle parallele. </s>
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3420 <s>Di modo che la linea del taglio, intendasi come altri vuole, non potrà esser
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parents:
diff changeset
3421 linea del taglio, se non costituisce varie maniere d’angoli, che una linea
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3422 retta non può già mai tagliare un’altra che non formi qualche maniera
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3423 d’angolo; che l’angolo (sì come può esser noto per la geometria) non si può
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3424 formar senza contatto o inclinatione, o declinatione, né l’inclinatione, o
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3425 la declinatione, il contatto può farsi già mai senza qualche tagliamento di
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3426 linea e così per opposito.</s>
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3427 </p>
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3428 <p type="main">
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3429 <s>Adunque la linea del taglio si produce nel medesimo tempo, che si formano gli
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3430 angoli.</s>
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3431 </p>
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3432 <p type="main">
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3433 <s>Oltre acciò per la linea del taglio si può prendere la linea del piano, o
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parents:
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3434 della terra; perciochè in essa si terminano tutte le perpendicolari che
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diff changeset
3435 nascono dagli angoli delle figure geometriche, o delle piante de’ corpi, che
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3436 si hanno a levare in prospettiva.</s>
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3437 </p>
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3438 <p type="main">
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diff changeset
3439 <s>Onde la detta linea venendo tagliata diversamente, porge le misure di ciò che
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diff changeset
3440 si dee fare nel piano iscorciato. </s>
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diff changeset
3441 <s>Si adoperano gli angoli nel trasportamento delle figure geometriche; perciò
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
3442 che elle non si potran trasportare se con le linee e co’ raggi visuali non
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diff changeset
3443 si truovano gli angoli delle figure, o le linee perpendicolari, che nascono
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diff changeset
3444 da essi, il che depende dalla ragione ottica (sì come si può vedere
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diff changeset
3445 osservando le regole, con le quali si trasportano: e la suppositione prima e
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diff changeset
3446 seconda della Prosp. di Eucl.).</s>
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diff changeset
3447 </p>
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diff changeset
3448 <p type="main">
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diff changeset
3449 <s>Negli intersegamenti delle linee col mezzo de’ quali si dispongano e
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parents:
diff changeset
3450 compartiscano i piani digradati, e le figure sopra essi, e vi si ergano i
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parents:
diff changeset
3451 corpi regolari, o irregolari, apparisce espressamente l’uso degli angoli;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3452 perciochè non si <pb pagenum="folios 62v-63r"></pb>possan già mai formar gli
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diff changeset
3453 intersegamenti senza costituir varie specie d’angoli secondo la diversa
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diff changeset
3454 applicatione delle regole di Prosp..</s>
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diff changeset
3455 </p>
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diff changeset
3456 <p type="main">
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diff changeset
3457 <s>Né le cose vere con le finte si possan bene accordare insieme senza servirsi
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3458 degli angoli, perciochè se riguardiamo i due piani d’una scena, l’uno finto
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parents:
diff changeset
3459 che si ritruova nella maestà, e l’altro vero, che è ‘l piano del palco, dove
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3460 si posano le cose sode e dove si rappresenta l’attion della commedia vi
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parents:
diff changeset
3461 vedremo l’intersegamento di due triangoli, mentre i lati dell’uno son
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parents:
diff changeset
3462 tagliati da’ lati dell’altro e ‘l tagliamento farsi ad angoli ottusi;
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parents:
diff changeset
3463 perciochè ‘l fine di questo intersegamento è che i due piani si convertano
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parents:
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3464 in uno stesso piano, né ciò possono fare se non si tagliano ad angoli ottusi
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diff changeset
3465 i lati di amendue ‘triangoli, come si vede in questa figura qui
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diff changeset
3466 appresso.</s>
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diff changeset
3467 </p>
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diff changeset
3468 <p type="main">
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parents:
diff changeset
3469 <s>Che acciochè di due triangoli se ne faccia uno bisogna che ‘quattro lati di
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3470 essi divenghino due, e non posson già mai i quttro lati de’ due triangoli
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3471 nell’apparenza diventar due lati d’un sol triangolo, se tagliandosi insieme
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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3472 non costituiscono angoli ottusi; perciochè a voler di due lati far uno in
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parents:
diff changeset
3473 apparenza,bisogna far che, tagliandosi, formino una maniera d’angoli che
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parents:
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3474 opposti alla vista nostra di lontano si perdano, e questi non sono altro che
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diff changeset
3475 gli ottusi, i quali vie più degli altri, per lor natura si accostano alla
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parents:
diff changeset
3476 linea retta, come altrove si è dimostrato.</s>
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diff changeset
3477 </p>
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diff changeset
3478 <p type="main">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3479 <s>Di modo che mentre si tagliano i lati de’ due triangoli formando angoli
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
3480 ottusi, i quali svaniscono dalla vista nostra, di maniera si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
3481 <lb></lb>//<lb></lb>uniscano, che di essi si forma una stessa linea.</s>
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diff changeset
3482 </p>
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diff changeset
3483 <p type="main">
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diff changeset
3484 <s>Onde venendo fatti due lati, i quattro de’ due triangoli, necessariamente
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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3485 diventeranno un solo, e così di due piani terminati da essi, se ne sarà
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3486 formato un piano solo, intendendo sempre in riguardo dell’apparenza; e
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parents:
diff changeset
3487 nell’esser loro sì come saranno quattro lati di due triangoli così saranno
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3488 due triangoli e per conseguenza due diversi piani come si potrebbe
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3489 dimostrare nella presente figura.</s>
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3490 </p>
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3491 <p type="main">
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3492 <s>Ma lasciamo queste considerationi alle teoriche della scenografia. </s>
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diff changeset
3493 <s>Nella stessa guisa si scuopre l’uso degli angoli nelle cose finte negli
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
3494 angoli della stanza; perciochè si costituiscano delle intersegation delle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3495 figure stesse, che vi si dipengano.</s>
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parents:
diff changeset
3496 </p>
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3497 <p type="main">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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3498 <s>Perciochè per le medesime ragioni vi si fa l’intersegatione ad angoli ottusi,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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3499 come si vede nel quadrolongo ABCD, al quale corrisponde nell’apparenza
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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3500 AEBCFD.</s>
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diff changeset
3501 </p>
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3502 <p type="main">
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3503 <s>Perciochè perdendosi l’angolo AEB e l’angolo CFD la linea AEB si fa
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3504 equivalente alla linea AB e la CFD alla CD.</s>
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3505 </p>
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3506 <p type="main">
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3507 <s>Finalmente nella disposition de’ piani, e delle elevation delle faccie e de’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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3508 coprimenti si vede espressamente l’uso degli angoli; perciochè non si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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3509 possono iscorciare senza la figura triangolare, che si costituisce dalle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3510 linee orizzontali, le quali congiognendosi in un ponto formano angolo per lo
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3511 più ottuso, come si vede chiarissimamente nella pratica della Scenografia. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
3512 <s>E la ragione di ciò è fondata nella suppositione 8 e 9 e nel teorema 4 e 5
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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3513 della Prospettiva d’Euclide, che mentre i raggi visuali son più alti e gli
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3514 intervalli di essi, benchè uguali da lontano apparendo minori, e gli
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3515 obbietti vicini all’occhio essendo maggiori son cagione che necessariamente
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diff changeset
3516 <pb pagenum="folios 63v-64r"></pb>i detti raggi visuali e le linee dell&#039;
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3517 orizzonte costituiscono qualche maniera d’angoli, come può esser noto a
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3518 chiunque osserva la Prospettiva.</s>
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3519 </p>
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3520 <p type="main">
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3521 <s>E questo sia a bastanza per raccontar l’uso degli angoli nella prospettiva
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3522 Scenografica; perciochè tutto l’uso di essi per quanto s’è fin qui da me
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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3523 osservato consiste in questi quattro capi o in altri che ad essi si
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diff changeset
3524 riducano; che se altri con più diligenza di me osservando le ragioni e le
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parents:
diff changeset
3525 pratiche di questa parte di Prospettiva trovarà altri usi d’angoli in essa
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parents:
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3526 ve l’aggionghi che a me è bastato solamente trovarne tanti quanti fussero
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3527 bastevoli a pruovar la mia ‘ntentione.</s>
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3528 </p>
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3529 <p type="main">
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3530 <s></s>
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3531 </p>
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3532 <p type="main">
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diff changeset
3533 <s></s>
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diff changeset
3534 </p>
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diff changeset
3535 </chap>
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diff changeset
3536 <chap>
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diff changeset
3537 <p type="head">
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3538 <s>Nelle Meccaniche</s>
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3539 </p>
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diff changeset
3540 <p type="head">
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diff changeset
3541 <s>Cap. 17</s>
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diff changeset
3542 </p>
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diff changeset
3543 <p type="main">
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diff changeset
3544 <s>Tutta la somma delle Meccaniche si riduce alla lieva, alla bilanca ed alla
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3545 statera, sì come apertamente potrà vedere qualunque con diligenza osservarà
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diff changeset
3546 Le Questioni Mec. d’Arist. e ‘n queste è manifesto l’uso degli angoli;
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diff changeset
3547 perciochè la lieva ( stromento col mezzo del quale qualunque movente, benchè
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diff changeset
3548 di poca virtù, muove, con gran facilità ogni peso, al cui movimento
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diff changeset
3549 naturalmente si richiedano più moventi) non muove peso alcuno se posata nel
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diff changeset
3550 piano della terra non forma angoli disuguali e se fermata nel centro che è
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diff changeset
3551 ‘l sotto lieva in esso non costituisce angoli uguali o simili: finalmente se
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diff changeset
3552 ella con la sua estremità formando una portion di circonferenza in essa non
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diff changeset
3553 costituisce angoli retti sferali, come si può vedere in A. B. Perciochè nel
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diff changeset
3554 movimento della lieva si presuppongano cinque cose, come si può ritrar da
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diff changeset
3555 Arist. nelle Mecc. <lb></lb>//<lb></lb>ciò sono el termine che sta immobile;
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3556 perciochè tutto quello che si muove si muove sopra qualche cosa immobile e
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diff changeset
3557 questo si è ‘l centro detto sotto lieva o quel che è posto in sua vece; la
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diff changeset
3558 linea della terra perciochè le cose gravi che si muovano dalla forza della
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diff changeset
3559 lieva si muovano per lo spatio che è nella superficie della terra; la lieva
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diff changeset
3560 stessa che è lo stromento che si adatta al movimento e supplisce la forza
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diff changeset
3561 manchevole nel movente; la linea che scaturisce da centro che rappresenta la
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diff changeset
3562 stessa linea; le circonferenze formate dalle due stremità della lieva. </s>
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3563 <s>Vi si aggiogne il termine mobile che sono le stremità della lieva: el movente
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diff changeset
3564 congionto con la lieva, che prende la virtù di molti moventi insieme uniti. </s>
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diff changeset
3565 <s>E per ripigliar da capo tutte le sopra dette cose bisogna dire che posta la
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parents:
diff changeset
3566 linea della terra e ‘l termine immobile sul quale si regge e si muove la
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diff changeset
3567 lieva e la linea della stessa lieva necessariamente bisogna che passando pel
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3568 detto punto e terminando nella linea della terra formi da una banda un
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diff changeset
3569 angolo ottuso e dall’altra un acuto; perciochè la natura della lieva non è
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3570 d’esser secondo la linea perpendicolare ma secondo la inclinata alla
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parents:
diff changeset
3571 giacente del piano che tosto che la lieva è fatta perpendicolare sul piano
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3572 della terra ha finito insieme col suo corso tutta la sua forza; sì come
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diff changeset
3573 movendosi la linea A.D. fermata ne centro C. infine al B. di maniera che si
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3574 converta nella B.D. perpendicolare, segue di muovere il peso C.F. ma se più
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3575 oltre si muovesse, non havrebbe<pb pagenum="folios 64v-65r"></pb>più alcuna
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parents:
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3576 forza di muovere il peso non andando incontra ‘l peso e non spegnendolo; ma
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3577 mutando la lieva e fermandola nel centro C. e sopra il piano della terra sul
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3578 ponto F. e muovendosi F. E. verso G.. Adunque se tutta la forza della linea
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3579 consiste nello spegnere il peso bisogna necessariamente che ogni volta che
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3580 la linea è fatta perpendicolare non habbia più forza. </s>
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3581 <s>Adunque l’angolo della costitution della lieva non è ‘l retto, ma l’ottuso e
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3582 l’acuto. </s>
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3583 <s>Così anchora per la medesima ragione tosto che la lieva A. C. è pervenuta
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3584 alla linea F. E. giacente che è la linea della terra perde tutta la forza
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3585 che haveva nell’alzare ‘l peso D. mentre si muove sopra la sottolieva
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3586 B.perciochè all’hora insieme con la linea del peso forma l’angolo retto. </s>
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3587 <s>Overamente si aggionga che all’hora accostandosi alla F. E. non forma più
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3588 angolo e perciò perde la forza di muovere. </s>
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3589 <s>Adunque bisogna che la lieva si costituisca non ad angoli retti, ma acuti ed
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3590 ottusi. </s>
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3591 <s>Oltre acciò non si può negar che nel collocar la lieva non formino gli
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3592 angoli; che ella non è altro che una linea materiale posata nel piano o
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diff changeset
3593 inclinante verso esso o da esso declinante. </s>
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3594 <s>E se la lieva è uno stromento manuale che mentre si muove di movimento
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3595 circolare forma più linee, che si partano dal centro immobile e terminano
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3596 nella circonferenza formata da lui, perché non si dee credere che nel centro
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diff changeset
3597 costituisca più angoli ed ottusi ed acuti e nella circonferenza più angoli
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3598 retti? Ciò si persuade a bastanza dalla ragione e dalla sperienza, come
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diff changeset
3599 apertamente vedrà chi osser <lb></lb>//<lb></lb>varà ’l movimento della lieva
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3600 espresso in questo essempio: e come si può veder sensatamente nelle taglie e
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3601 nelle ruote: Senza che quel che si muove proportionatamente sopra la linea
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3602 retta si muove secondo ‘l diametro e quanti sono i movimenti tanti i
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3603 diametri i quali tagliandosi nel centro formano angoli. </s>
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3604 <s>Finalmente la bilancia se riguardiamo alla postura dell’equilibrio, essendo
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3605 fermata nel suo centro con la lenguetta che sopra esso si eleva dentro
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3606 all’aco la qual da Greci si appella Carcone e da’ Latini Correttore et
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diff changeset
3607 Virgula, forma angoli retti insieme con esso pervenendo nella stessa linea
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3608 perpendicolare in quella guisa che fa l’Archipendolo de’ muratori, mentre ‘l
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3609 filo s’affronta nelle due tacche come si vede qui appresso nell’essempio e
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3610 dalla cognition di quest’ angoli retti subbito si viene alla notitia
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diff changeset
3611 dell’equilibrio. </s>
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3612 <s>Ma osservata la bilancia fuor di questa positura, in qualunque modo sia,
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3613 essendo fermata nel medesimo centro, si vedrà che la detta lenguetta insieme
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3614 co l’aco non si accorda non havendo commune la perpendicolare e perciò non
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parents:
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3615 forma più l’angolo retto, ma l’acuto e l’ottuso. </s>
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3616 <s>Di modo che tosto veduto un angolo diverso dal retto e non esser commune la
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3617 perpendicolare si conosce la bilancia esser fuor dell’equilibrio onde si
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diff changeset
3618 scorge la differenza del peso delle cose poste in amendue le bilance. </s>
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3619 <s>Ma nella statera è necessario osservare gli angoli retti nell’aggiustamento
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parents:
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3620 della lenguetta con l’aco e gli angoli acuti per conoscer l’alzamento del
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parents:
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3621 contrapeso e la quantità della cosa che si pesa con la statera, come qui si
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parents:
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3622 vede. e questo <pb pagenum="folios 65v-66r"></pb>è quanto m’occoreva dire
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parents:
diff changeset
3623 intorno all’uso degli angoli nelle Meccaniche lassando molt’altre cose da
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diff changeset
3624 dirti che pienamente si trattano da Arist. e da altri autori moderni a’
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diff changeset
3625 quali mi rimetto, che in questo luogo m’è paruto più convenevole il
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parents:
diff changeset
3626 considerar le parti principali della scienza meccanica che tutte le cose ad
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parents:
diff changeset
3627 essa appartenenti, non essendo questo luogo convenevole.</s>
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diff changeset
3628 </p>
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parents:
diff changeset
3629 <p type="main">
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parents:
diff changeset
3630 <s></s>
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parents:
diff changeset
3631 </p>
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parents:
diff changeset
3632 </chap>
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parents:
diff changeset
3633 <chap>
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parents:
diff changeset
3634 <p type="head">
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parents:
diff changeset
3635 <s>Nell’Architettura e particolarmente nella ornata e nella militare</s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3636 </p>
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parents:
diff changeset
3637 <p type="head">
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parents:
diff changeset
3638 <s>Cap. 18</s>
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parents:
diff changeset
3639 </p>
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parents:
diff changeset
3640 <p type="main">
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parents:
diff changeset
3641 <s>Non è dubbio alcuno che l’Architettura generalmente considerata e
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3642 specialmente l’Architettura ornata e la militare ha principalmente
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3643 dependenza dalla Geometria; perciochè ciascuna di queste prende i punti, le
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3644 linee e le superficie diversamente figurate e corpi altresì variamente
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parents:
diff changeset
3645 terminati e li adatta alla materia trattabile; che nella pura Geometria si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3646 considerano separati da ogni materia. </s>
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parents:
diff changeset
3647 <s>Però, sì come nella Geometria pura per cagion delle dette cose si scorge
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parents:
diff changeset
3648 l’uso degli angoli, così col mezzo delle medesime si potrà comprender
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parents:
diff changeset
3649 nell’Architettura ornata e militare. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3650 <s>E per dimostrare il detto uso in ciascuna di queste proporrò primieramente
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parents:
diff changeset
3651 tutte le parti dell’Architettura ornata nelle quali apparisce espressamente
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3652 l’uso degli angoli. </s>
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parents:
diff changeset
3653 <s>Perciochè essendo più propria della pace che della guerra par che
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3654 ragionevolmente le convenga il primo luogo: e alla militare convenendo molto
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3655 più alla guerra che alla pace con ragione si adatta il secondo luogo; che la
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3656 pace in quanto che è ‘l conservamento delle cose ed in quanto è fine della
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parents:
diff changeset
3657 guerra merita d’esser tenuta più degna, e perciò l’Architettura
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3658 <lb></lb>//<lb></lb>ornata appartenendo alla pace doverà ottenere il primo luogo in
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parents:
diff changeset
3659 questo nostro discorso. </s>
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parents:
diff changeset
3660 </p>
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parents:
diff changeset
3661 <p type="main">
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parents:
diff changeset
3662 <s>Mostriamo adunque prima le parti nelle quali consiste l’uso degli angoli
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3663 appresso l’ornata e consideriamole a parte a parte. </s>
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parents:
diff changeset
3664 <s>L’uso degli angoli in quest’Architettura si comprende nelle figure angolari
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3665 di varie maniere, o sieno regolari o irregolari, le quali si adoperano nel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3666 formar le piante delle fabbriche e nel compartirle.</s>
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parents:
diff changeset
3667 </p>
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parents:
diff changeset
3668 <p type="main">
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parents:
diff changeset
3669 <s>Nel compartimento delle facciate degli edifitij.</s>
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parents:
diff changeset
3670 </p>
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parents:
diff changeset
3671 <p type="main">
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parents:
diff changeset
3672 <s>Nel ridurre in piano le cose.</s>
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parents:
diff changeset
3673 </p>
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parents:
diff changeset
3674 <p type="main">
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parents:
diff changeset
3675 <s>Nel tirar a piombo e dirittamente le muraglie.</s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3676 </p>
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parents:
diff changeset
3677 <p type="main">
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parents:
diff changeset
3678 <s>Nell’adoperar la squadra e l’archipendolo.</s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3679 </p>
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parents:
diff changeset
3680 <p type="main">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3681 <s>Nel formar i disegni delle fabbriche.</s>
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parents:
diff changeset
3682 </p>
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parents:
diff changeset
3683 <p type="main">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3684 <s>Ne’compartimenti della superficie de’ piani e delle città.</s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3685 </p>
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parents:
diff changeset
3686 <p type="main">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3687 <s>Ne’ compartimenti delle città si vede apertissimamente l’uso degli angoli;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3688 perciochè eletta già l’area in cui si debba fabbricar la città (supposto che
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3689 si habbia da edificar tutta in un tempo e non in più tempi successivi ed in
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parents:
diff changeset
3690 più diversi aggiognimenti, come spesso accade) e collocato in mezzo ad essa
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parents:
diff changeset
3691 uno stromento di marmo da Vitruvio appellato Amusio, nel mezzo del quale sia
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3692 collocato lo Gnomone e secondo l’osservation dell’ombra tante hore avanti
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3693 quante doppo mezzogiorno, ritrovata la linea meridiana e quindi anchora
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3694 ritrovati tutti i venti secondo la pratica insegnataci da Vitruvio si divide
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3695 tutta la sua circonferenza overo (come modernamente si costuma) con l’uso
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3696 della Bossola detta Agucchia, secondo ‘l numero de’ venti, cioè in XXIIII
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3697 parti, e ‘n tutta l’area si disegnano due quadrati perfetti che
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parents:
diff changeset
3698 s’interseghino insieme i quali cos <pb pagenum="folios 66v-67r"></pb>tituiscano
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parents:
diff changeset
3699 gli VIII venti: così anchora vi si formano due ottangoli i quali pur’ancho
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parents:
diff changeset
3700 si taglino fra loro e da amendue si divide la circonferenza in sedici parti
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parents:
diff changeset
3701 di maniera che ogni quarta vien divisa in quattro parti uguali. </s>
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parents:
diff changeset
3702 <s>Ma di nuovo si divide in tre parti tali che tutta essendo divisa ugualmente
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parents:
diff changeset
3703 costituisce XXIIII venti. </s>
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parents:
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3704 <s>Di maniera che tagliandosi insieme i lati delle suddette figure si
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parents:
diff changeset
3705 costituiranno angoli diversi e disuguali come ottusi e acuti, più acuti o
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parents:
diff changeset
3706 meno acuti, interni ed esterni ed opposti, come si potrebbe render chiaro
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parents:
diff changeset
3707 per la figura. </s>
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parents:
diff changeset
3708 <s>E secondo la disposition di queste linee prodotte da’ venti e secondo ‘l
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parents:
diff changeset
3709 tagliamento de’ lati delle dette figure si forma tutto ‘l compartimento
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parents:
diff changeset
3710 della città, cioè le piazze, le strade, i vicoli, gli angiporti, l’isole, i
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parents:
diff changeset
3711 borghi e ‘siti de’ tempij e d’altri edificij pubblici, e delle porte e delle
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parents:
diff changeset
3712 torri. </s>
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parents:
diff changeset
3713 <s>Onde essendo queste parti della città formate dentro alle linee de’ venti e
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parents:
diff changeset
3714 dentro gl’intersegamenti loro, da’ quali si costituiscano gli angoli,
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parents:
diff changeset
3715 haveranno necessariamente il sito loro determinato dentro qualche maniera
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parents:
diff changeset
3716 d’angoli; per la qual cosa ci si manifesta l’uso degli angoli nel fabbricar
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3717 le città, come molto meglio ciascuno si potrà certificare che osservi Cesare
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parents:
diff changeset
3718 Cesariano Milanese nel Comm. di Vitruvio; là dove e con parole e con disegno
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3719 dimostra eccellentissimamente il compartimento della città.</s>
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parents:
diff changeset
3720 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3721 <p type="main">
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parents:
diff changeset
3722 <s>Nel compartir le superficie de’ piani, cioè di pavimenti degli edificij e
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3723 delle piazze e di giardini è necessario servirsi degli angoli; perciochè
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3724 sieno le dette superficie di qualunque figura per for <lb></lb>//<lb></lb>mar in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3725 esse qualsivoglia maniera di compartimento bisogna risolverla in più figure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3726 o della medesima specie o di specie diversa, regolari o irregolari; né si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3727 possan formarvi le dette figure senza costituir varie ragioni d’angoli. </s>
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parents:
diff changeset
3728 <s>Ed in questo si manifesta l’uso loro.</s>
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parents:
diff changeset
3729 </p>
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parents:
diff changeset
3730 <p type="main">
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parents:
diff changeset
3731 <s>I disegni delle fabbriche, o sieno di piante, o di elevationi non si possan
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3732 far senza servirsi degli angoli perciochè si formano tirando prima una linea
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3733 retta giacente e di poi una perpendicolare nel mezzo dello spatio, cadente
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3734 sopra essa e quindi altre linee parallele piane e di poi altre
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3735 perpendicolari parimente parallele alla prima perpendicolare le quali
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parents:
diff changeset
3736 cadendo sopra le linee piane per la .12. definitione e per la .13. e .14.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3737 prop. del primo di Euclide, non posson non formare angoli retti o uguali a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3738 più retti in fra di loro conseguenti. </s>
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parents:
diff changeset
3739 <s>E questo si è l’uso degli angoli che in questa parte necessariamente si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3740 richiede; perciochè in tale operatione si adoperano le linee piane e le
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casties
parents:
diff changeset
3741 linee a piombo, con le quali naturalmente si accompagnano gli angoli
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3742 retti.</s>
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parents:
diff changeset
3743 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3744 <p type="main">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3745 <s>Nell’adoperar la squadra si vede manifestamente l’uso degli angoli perciochè
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3746 la squadra non è altro che l’angolo retto congionto con la materia; onde è
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3747 che dagli Architetti l’angolo retto è detto angolo a squadra. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3748 <s>Perciò che la squadra si compon di due linee rette una cadente sopra l’altra
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3749 ad angoli retti, e così anchora l’angolo retto come è manifesto per 12. def.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3750 già citata. </s>
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casties
parents:
diff changeset
3751 <s>E nell’uso dell’archipendolo si adopra l’angolo; perciochè all’hora sta
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3752 diritto e giusto e (come si dice) a livella, quando ‘l filo e ‘l piombino,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3753 detto linea della fiducia, cade perpendicolarmente e forma nella traversa da
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3754 ogni banda angoli retti, dividen <pb pagenum="folios 67v-68r"></pb>dola in due
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3755 parti uguali, come ancho dividendo l’angolo in parti uguali, come è
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3756 manifesto per la nona, per la decima, per l’undicesima e per la dodicesima
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3757 propositione del primo di Euclide, onde nasce la fabbrica di tale
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3758 stromento.</s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3759 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3760 <p type="main">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3761 <s>Nel tirar a piombo e addirizzar le muraglie si vede l’uso degli angoli;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3762 perciochè la linea del piombo sopra quella del piano o sopra qualunque altra
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3763 giacente o sopra riga o regolo forma angoli retti onde quando gli angoli non
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3764 appariscano retti all’hora si conosce la cosa non esser in piano, così
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3765 anchora quando ‘l filo che guida l’elevation d’una muraglia non forma
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3766 angoliretti con quello del piano, dà segno che la fabbrica pende; e
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3767 riducendola col mezzo del filo si addirizza e si accommoda a piombo.</s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3768 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3769 <p type="main">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3770 <s>Nel ridurre in piano ciascuna cosa bisogna adoperare gli angoli; perciochè si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3771 usa l’archipendolo col mezzo del quale all’hora si conosce la cosa esser in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3772 piano, quando la linea della fiducia tagliando ugualmente la linea
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3773 trasverale dello stromento forma da ogni parte angoli retti.</s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3774 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3775 <p type="main">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3776 <s>Ancho nel compartir le facciate delle fabbriche concorre l’uso degli angoli;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3777 perciochè primieramente non è facciata di qualunque sia fabbrica che non sia
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3778 di qualche figura angolare, levatane la circolare e l’ornata, la quale
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3779 anchora non può non servirsi degli angoli acciochè habbia posamento nel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3780 terreno. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3781 <s>E stando sopra ‘l piano ciascuna fabbrica perpendicolarmente bisogna che
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3782 formi sopra esso angoli retti, come si potrebbe pruovare per la diciottesima
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3783 dell’undecimo d’Euclide. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3784 <s>Ed havendo a riempier d’ornamenti tutto lo spatio delle facciate è necessario
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3785 risolverle in molte figure, le quali non <lb></lb>//<lb></lb>saranno senza qualche
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3786 maniera d’angoli; perciochè tutti i vani, come porte, finestre, quadri,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3787 nicchie, intervali ed altre cose simili non si possan formar se non con gli
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3788 angoli.</s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3789 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3790 <p type="main">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3791 <s>Finalmente nelle figure angolari, le quali concorreno nel componimento di
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3792 tutte le fabbriche si vede espressamente l’uso degli angoli; perciochè di
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3793 qualunque figura sieno è necessario per compartir tutto il loro spatio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3794 formarvi molte figure angolari secondo che richiede la varia qualità loro. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3795 <s>Di qui adunque in tutte queste cose appartenenti all’Architettura Ornata si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3796 manifesta l’uso degli angoli, sì come habbiamo accennato. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3797 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3798 <p type="main">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3799 <s>Resta hora a considerare in ciò che sia collocato l’uso degli angoli nella
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3800 Militare.</s>
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casties
parents:
diff changeset
3801 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3802 <p type="main">
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parents:
diff changeset
3803 <s>L’uso degli angoli nella Fortificatione si dimostra in più maniere; perciochè
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3804 principalmente sopra gli angoli di qualunque figura regolare o irregolare si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3805 fonda tutta la fortification delle fortezze e delle città, perciochè
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3806 l’angolo per sua strettezza essendo debile in maniera che non può molto
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3807 resister alla forza e all’impeto de’ colpi delle batterie, ha bisogno
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3808 d’esser fortificato: e perché lo spatio che si racchiude in ciascuno angolo
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3809 è troppo breve, onde non è atto a dar commodità a’ difensori ed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3810 all’artigliarie che vi si richiedano per difesa del luogo (che la fortezza
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3811 degli angoli consiste nel poter haver le piazze larghe) né alle ritirate,
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parents:
diff changeset
3812 che vi fusse bisogno di fare, però fa di mestiero allargarlo. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3813 <s>Onde non è maraviglia se Bonaiuto Lorini dica che ogni maniera di
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casties
parents:
diff changeset
3814 fortificatione si formi sopra gli angoli alla qual sentenza si
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parents:
diff changeset
3815 sottoscriverebbe ciascuno <pb pagenum="folios 68v-69r"></pb>Ingegnero; perciochè
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parents:
diff changeset
3816 questo dalla sperienza si conferma. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3817 <s>Oltre acciò quest’uso si manifesta nel formar i disegni della fortificatione;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3818 poiché si formano col mezzo delle linee delle difese e co’ tiri
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3819 dell’Artigliarie che si descrivano mentre si formano i profili delle
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parents:
diff changeset
3820 fortezze. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3821 <s>Le qua’ linee e qua’ tiri si tagliano in fra loro e formano varie maniere
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3822 d’angoli. </s>
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parents:
diff changeset
3823 <s>Finalmente apparisce quest’uso nel levar le piante delle fortezze o delle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3824 città o con istromenti o senza ritrahendo al costume de’pittori e de’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3825 disegnatori; perciochè non si possan levar tali piante senza trovar prima
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3826 gli angoli loro. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3827 <s>Ma se da alcuno si ricercasse per avventura quali angoli sieno più
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3828 frequentati nel fortificare e quali ragionevolmente convenghino alla buona
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parents:
diff changeset
3829 Architettura militare, si potrebbe risponder esser gli angoli ottusi,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3830 intendendo però per il più che di rado si serve degli acuti e de’retti. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3831 <s>E in questo luogo intendiamo degli angoli de’ ricinti delle città e delle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3832 fortezze, sopra ‘ quali si fonda la fortificatione; perciochè gli angoli
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3833 usati ne’ baluardi sogliono essere acuti, benchè spesso sieno ottusi; che si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3834 variano non solamente per cagion della figura di più o di men lati, ma per
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3835 la varietà del luogo de’ punti delle difese, preso o nel mezzo della
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parents:
diff changeset
3836 cortina, o a’ due terzi, o nel collo del baluardo. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3837 <s>Potermmo aggiognere che l’uso degli angoli si manifesta con le linee che
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3838 formano i fianchi del baluardo; perciochè tagliano la linea della cortina ad
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3839 angoli retti, il che si vede chiaramente; perciochè queste linee o si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3840 formano <lb></lb>//<lb></lb>dalla squadra o da quella operatione del compasso per la
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3841 quale si produce la linea perpendicolare. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3842 <s>E questo basti haver detto dell’uso degli angoli nell’Architettura militare;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3843 che in essa non hanno altri usi che questi e se pur si hanno degli altri,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3844 almen non son così principali.</s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3845 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3846 <p type="main">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3847 <s></s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3848 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3849 </chap>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3850 <chap>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3851 <p type="head">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3852 <s>Nell’arte militare</s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3853 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3854 <p type="head">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3855 <s>Cap. 19</s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3856 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3857 <p type="main">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3858 <s>Fra tutte le cose che al’arte della guerra appartengono una ve n’è nella
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3859 quale apparisce l’uso degli angoli e questa si è ‘l modo di accommodar in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3860 campo i battaglioni degli eserciti. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3861 <s>Ma in essa non si vede tale uso per sé, ma per accidente, cioè per mezzo
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3862 delle figure costituite da’ soldati; perciochè con essi si formano varie
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3863 maniere di figure angolari, ciò sono quadrate, quadre longhe, triangolari,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3864 pentagone; ma inregolari, che da altri son dette triangolari; ma per
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3865 essergli levati due angoli appresso alla base con due tagliamenti, vi si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3866 costituiscano quattro angoli onde insieme con l’angolo superiore del
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3867 triangolo risultano cinque angolie la figura di cinque lati: a lunetta e a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3868 forbice, tutte figure simplici e angolari. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3869 <s>Altre sono figure composte o con più figure quadre congionte insieme con una
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3870 parte di lunetta, sì come è la figura simigliante allo scorpione: overo un
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3871 quadro congionto con due lunette, posto in fra ‘due convessi di esse, come
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3872 si costuma colle picche: o un quadro perfetto con due braccia da una fronte,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3873 che son due metà di due lunette solito farci con le picche: o con quattro
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3874 quadri perfetti accomodati a modo di croce, nel cui mezzo si pongano le
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3875 bandiere, le lance spezzate e <pb pagenum="folios 69v-70r"></pb>gli huomini
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3876 grandi guardati in ogni angolo dalle bombarde: similmente con tre lunette,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3877 le cui fronti si formano da’ cannoni loro, di modo che in mezzo rimanga uno
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3878 spatio triangolare di linee curve, dove sieno in mezzo le insegne e le
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3879 persone principali guardate da ogn’angolo con l’artigliaria: o con una
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3880 figura triangolare accompagnata da ogni banda da un’ala di soldati fatta con
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3881 una portione di lunetta, sì che amedue l’ale servono per braccia; perciochè
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3882 la base di tal figura è una lunetta, che avanzando da ogni banda ugualmente,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3883 fa fronte verso l’angolo superiore, che è la fronte principale del
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3884 battaglione, come si suol fare con le picche: Finalmente con due romboidi e
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3885 con un rombo, che le congionga in modo che formino la lettera V e col
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3886 rimanente fatto in forma di lunetta stesa infine alle romboidi, la quale
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3887 rassembri un mezzo circo, nel mezzo del quale si racchiuda un quadro, figura
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3888 che per la simiglianza si domanda a forbice: e con altre figure composte le
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3889 quali contengono in sé diverse specie d’angoli. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3890 <s>Di modo che questi battaglioni sieno di figure così simplici, come di
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3891 composte, non possano già mai formarsi senza servirsi degli angoli come a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3892 chiunque considera bene qualsivoglia figura di battaglione, ansi le file
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3893 stesse de’ soldati in fra loro (che queste formano que’ medesimi angoli che
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3894 le figure formate da loro) apparirà chiarissimo l’uso degli angoli.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3895 <lb></lb>//<lb></lb>Ma quel che appartiene al modo di ordinar le file secondo i
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3896 numeri determinati, acciò produchino le figure, che s’intende di formare, si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3897 lassarà considerare a que’ che esattamente insegnano l’arte militare e
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3898 specialmente il modo di ordinar i battaglioni e d’accampar gli eserciti alla
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3899 campagna; onde rimettendomi agli autori che ne trattano, e specialmente a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3900 Girolamo Maggi e all’autore del libro intitolato Il Vallo e a tutti quelli
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parents:
diff changeset
3901 che n’hanno scritto, porrò fine a questo presente capitolo.</s>
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parents:
diff changeset
3902 </p>
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parents:
diff changeset
3903 <p type="main">
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parents:
diff changeset
3904 <s></s>
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casties
parents:
diff changeset
3905 </p>
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parents:
diff changeset
3906 </chap>
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parents:
diff changeset
3907 <chap>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3908 <p type="head">
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parents:
diff changeset
3909 <s>Nella Navigatoria</s>
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parents:
diff changeset
3910 </p>
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parents:
diff changeset
3911 <p type="head">
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parents:
diff changeset
3912 <s>Cap. 20</s>
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parents:
diff changeset
3913 </p>
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parents:
diff changeset
3914 <p type="main">
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parents:
diff changeset
3915 <s>Non è dubbio alcuno che chi riguarda bene l’arte del navigare vedrà
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parents:
diff changeset
3916 chiarissimamente l’uso degli angoli per se stesso non le esser necessario;
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parents:
diff changeset
3917 ma risultare dalle cose necessarie a tale arte. </s>
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parents:
diff changeset
3918 <s>Perciochè dalla costitution de’ venti rispetto alla sfera del mondo e dalle
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parents:
diff changeset
3919 linee formate dal movimento de’ Navilij, alle quali corrisponde la carta da
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parents:
diff changeset
3920 navigare e la bossola, stromenti necessarissimi de’ naviganti, scaturisce
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3921 l’uso degli angoli; poiché considerinsi venti in quanto formano le quarte e
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parents:
diff changeset
3922 ‘n quanto costituiscono i rombi, sempre apparirà chiarissimo l’uso degli
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parents:
diff changeset
3923 angoli; perciochè sì le quarte e sì i rombi non si formano senza angoli,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3924 considerinsi rispetto al centro del mondo dove concorreno o rispetto alla
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parents:
diff changeset
3925 circonferenza che è tagliata ad angoli retti da ciascun vento; perciochè
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parents:
diff changeset
3926 ogni vento forma una linea non retta; ma curva e più tosto circolare secondo
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3927 <pb pagenum="folios 70v-71r"></pb>la quale si muove e questa linea taglia ad
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parents:
diff changeset
3928 angoli retti la circonferenza della sfera della terra e dell’acqua: E benchè
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3929 i venti nella carta da navigare costituischino le linee rette, con tutto ciò
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3930 non si considerano né s’adoprano come rette; ma come circolari; onde è che
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3931 la medesima quantità di miglia e di leghe che nella tondezza della terra e
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3932 dell’acqua insieme prese si dimostra, apparisce ancho in piano per la carta
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parents:
diff changeset
3933 da navigare, così in terra come in acqua (sì come si ritrahe dall’Arte del
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parents:
diff changeset
3934 navigare di Pietro da Medina, nel cap. 7 del 3° libro) segnandosi con leghe
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parents:
diff changeset
3935 e gradi la lontananza, che habbia qualsivoglia cosa nella tondezza del
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3936 mondo, senza scemarle parte alcuna e ciò non è inconveniente, perché a un
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3937 corpo ritondo si può conceder la medesima proportione nel piano, come si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3938 dimostra da Tolomeo nel Planisferio e da Giordano nel trattato del medesimo. </s>
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parents:
diff changeset
3939 <s>Ma come questa proportione si dimostri e per qual cagione le misure delle
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parents:
diff changeset
3940 distanze delle cose, così in piano, come in tondo, sieno le medesime non si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
3941 dee ragionar in questo luogo, bastando solamente dimostrare in che consista
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diff changeset
3942 l’uso degli angoli nella Navigatoria, e questo esser il tagliamento della
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parents:
diff changeset
3943 circonferenza della sfera, la costitution delle varie portioni di essa,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3944 essendo già divisa da’ venti in quarte e ‘n rombi. </s>
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parents:
diff changeset
3945 <s>E si avvertisca che si considerano gli angoli nella Navigatoria non in quanto
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diff changeset
3946 all’utilità, che a tale arte apportar possino, perciochè in essa non si fa
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parents:
diff changeset
3947 conto alcuno degli angoli; ma in quanto che ci si trovano <lb></lb>//<lb></lb>quelle
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parents:
diff changeset
3948 cose che alla Navigatoria sono di grandissimo giovamento; che non si
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parents:
diff changeset
3949 riguarda quali angoli formino i venti nella carta da navigare, o nella
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diff changeset
3950 circonferenza della terra e dell’acqua, ma si considera la positura loro,
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parents:
diff changeset
3951 acciò si possa vedere per qual vento sia meglio navigare sicuramente e con
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diff changeset
3952 prestezza. </s>
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diff changeset
3953 <s>E questo basti intorno al proposito della materia di questo capitolo.</s>
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3954 </p>
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parents:
diff changeset
3955 <p type="main">
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parents:
diff changeset
3956 <s></s>
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parents:
diff changeset
3957 </p>
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parents:
diff changeset
3958 </chap>
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diff changeset
3959 <chap>
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parents:
diff changeset
3960 <p type="head">
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parents:
diff changeset
3961 <s>Nell’agricoltura</s>
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parents:
diff changeset
3962 </p>
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parents:
diff changeset
3963 <p type="head">
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parents:
diff changeset
3964 <s>Cap. 21</s>
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parents:
diff changeset
3965 </p>
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parents:
diff changeset
3966 <p type="main">
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parents:
diff changeset
3967 <s>Non è arte alcuna che servendosi in qualche maniera de’ lineamenti o delle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3968 superficie in vario modo terminate non si serva ancho degli angoli. </s>
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parents:
diff changeset
3969 <s>Questo si vede apertissimamente non solo per le cose dette ma ancho per la
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3970 osservatione dell’agricoltura; perciochè, nel porre gli arboli con ordine,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3971 come nel far gli oliveti, le selve, le lame, e nel piantar le vigne e gli
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3972 anguillacci e finalmente ne’ compartimenti degli ordini, tale uso apparisce
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3973 chiarissimamente. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3974 <s>E che altro dimostrano gli ordini degli arboli piantati (come si dice) a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3975 Quincunce, de’ quali fece mention Cicerone nel dialogo Della vecchiezza, se
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3976 non l’uso degli angoli? Quest’ordine così appellato da Cicerone stesso si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3977 espone in tal guisa:</s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3978 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3979 <p type="main">
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parents:
diff changeset
3980 <s>Cum autem admiraretur Lysander et processitates arborum et disectos in
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parents:
diff changeset
3981 quincuncem ordines. </s>
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parents:
diff changeset
3982 </p>
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parents:
diff changeset
3983 <p type="main">
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parents:
diff changeset
3984 <s>Come ancho dicendo:</s>
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parents:
diff changeset
3985 </p>
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parents:
diff changeset
3986 <p type="main">
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parents:
diff changeset
3987 <s>Quid de pratorum viriditate, aut arborum ordinibus, aut vinearium olivetorium
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3988 ne specie dicam?</s>
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parents:
diff changeset
3989 </p>
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parents:
diff changeset
3990 <p type="main">
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parents:
diff changeset
3991 <s>E Quintiliano nel cap. 3° dell’8° libro:</s>
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parents:
diff changeset
3992 </p>
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parents:
diff changeset
3993 <p type="main">
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3994 <s>Quid illo quincunce speciosius qui in quacumque partem spectaveris vetus est?
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
3995 <pb pagenum="folios 71v-72r"></pb></s>
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3996 </p>
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parents:
diff changeset
3997 <p type="main">
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parents:
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3998 <s>Ma Ambrogio Calepino così ‘l dichiara:</s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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3999 </p>
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4000 <p type="main">
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4001 <s>Quincumque dicitur, cum ita disposite sunt arbores, ut bine cum tertia
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diff changeset
4002 proximi ordinis equali intervallo sibi opposita quo’ quo’ te vertis
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4003 quincuncialem referunt.</s>
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diff changeset
4004 </p>
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4005 <p type="main">
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diff changeset
4006 <s>Il quale ordine si dimostra con tale essempio sensato qui appresso. </s>
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diff changeset
4007 <s>Ma per intender meglio questa voce Quincunce si dee avvertire che appresso
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diff changeset
4008 gli antichi il numero cinque, come in se stesso e come numerante si mostra
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diff changeset
4009 con questa quinta lettera vocale V, come dice Valerio Probo nel lib. </s>
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diff changeset
4010 <s>De Notis Romanorum e questo si era ‘l segno dimostrativo di tal numero
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4011 quinario e considerato come numero numerato, cioè in quanto che è applicato
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diff changeset
4012 al peso d’alcuna cosa, che pesi cinque once, si dice Quineunx e Quincunce,
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diff changeset
4013 come si vede appresso Pietro Diacono “De minutijs”, onde e dalla quantità
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diff changeset
4014 del peso e dalla sua misura, che è ‘l numero cinque riceve ‘l nome Quincunce
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4015 e si nota col medesimo segno, sì come ‘l segno dimostrativo del numero
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diff changeset
4016 Denario era la decima lettera X consonante e sì come ‘l cinque V è la metà
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diff changeset
4017 del dieci X, così ‘l segno Quincunce V è la metà del segno denario X. Oltre
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4018 acciò si avverta che la figura del Quincunce si forma col mezzo
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4019 dell’intersegatione di due linee; perciochè, poste alcune piante con uguale
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diff changeset
4020 intervallo, a due a due, gli si oppone sempre la terza del secondo ordine,
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4021 come si vede <lb></lb>//<lb></lb>qui appresso, che alle piante AB si oppone la
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4022 pianta C e alle EC la D e alle CF la G e alle DG la H e mentre da A al G e
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diff changeset
4023 da B a D si tirano le linee, che si tagliano insieme nel punto C, si forma
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diff changeset
4024 ‘l Quincunce da ogni banda, come ACB. DCG. DCG. DHG. EAC. EDC. CBF. CGF e
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diff changeset
4025 però disse Ambrogio Calepino: Quòquò te vertas quincuncialem referunt. </s>
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4026 <s>E adunque ‘l Quincunce degli antichi un ordine usato nel piantar gli arboli e
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4027 le viti, nel quale corrispondendo dirittamente le file di essi ed
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4028 intersegandosi da ogni parte le linee loro, mostrano da ogni banda la figura
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4029 di questo segno V Quincunce, in quella stessa guisa che si vede in Roma
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diff changeset
4030 quell’edifitio vulgarmente appellato le “sette sale”, che anticamente era
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diff changeset
4031 una conserva d’acque, detta “castello dell’acque”. E perché quest’ordine non
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4032 può formar la detta figura senza costituir angoli, quindi è che possiamo
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4033 dire che nel formarsi ‘l piantamento a Quincunce, cioè a mandorla, si
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4034 adoperano gli angoli. </s>
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4035 <s>Perciochè non è altro l’ordine a quincunce, che ‘l tirar due linee, le quali
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4036 si taglino in fra loro in quella stessa guisa che si dimostra da Euclide
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4037 nella quindicesima propositione del primo libro: e nel tirar dette linee si
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4038 costituiscano quattro angoli che che da lui si appellano “angoli ad
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4039 verticem”. I quali, secondo che egli dimostra sono fra loro uguali. </s>
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4040 <s>E che ciò sia vero rimiriamo con diligenza il Quincunce e restaremo accertati
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4041 esser ‘l tagliamento di due linee rette AB. CD. nel ponto E e delle GH. GB.
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4042 nel ponto F., e di più vi potremo osservare il Quincunce non esser altro che
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4043 l’angolo AEC. CFG. DEB. BFH. AED. CFB. CEB. GFH, onde si può concludere che
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4044 ‘l piantare a Quincunce non sia altro che’l formar gli angoli ne’
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4045 tagliamenti delle linee rette in fra loro opposti direttamente e per
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4046 diametro uguali, cioè AEC. all’angolo DEB. AED <pb pagenum="folios 72v-73r"></pb>a CEB. e così degli altri. </s>
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4047 <s>Che ‘l Quincunce AEC è l’angolo stesso AEC. così ‘l DEB. Si potrebbe ancho,
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4048 che nel Quincunce sia l’uso degli angoli esterni, interni e opposti formati
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4049 dal tagliamento delle parallele sopra le quali cada una linea retta, in
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4050 quella maniera che si dimostra da Euclide nella ventinovesima del primo, la
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4051 qual cosa si vede apertamente nell’osservar la figura dell’ordine a
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diff changeset
4052 Quincunce, la quale si compone con linee parallele, le quali si tagliano fra
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4053 loro e nel taglio costituiscono i detti angoli come si vede qui appresso e
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4054 come vediamo nelle ferrate delle finestre o nelle grate delle prigioni o de’
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4055 paralatorij delle monache. </s>
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parents:
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4056 <s>Si vede oltre acciò l’uso degli angoli ne’ compartimenti de’ campi, fatti per
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parents:
diff changeset
4057 distribuir a ciascuna parte di terreno la sementa ordinata per succession di
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parents:
diff changeset
4058 tempo e per iscambiamento di seme; perciochè mentre si dividano i campi in
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parents:
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4059 più parti si risolvano in molte e diverse figure regolari, secondo che
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parents:
diff changeset
4060 comporta tutta la disposition della figura del campo intero. </s>
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4061 <s>Si può ancho vedere nelle varie positure delle fosse e de’ solchi e degli
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parents:
diff changeset
4062 acquedotti che si fanno per cavar l’acque de’ campi; perciochè ne’ piani il
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4063 più delle volte si fano ad angoli retti nella estremità di essi: e nelle
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parents:
diff changeset
4064 coste e nelle colline si fanno terminanti nel fine e nel principio del campo
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4065 ad angoli disuguali, cioè ad angoli ottusi ed acuti per dar la pendina e lo
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4066 scolamento più facile e più sicuro all’acqua e con meno violenza e con
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parents:
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4067 maggior fermezza di terreno. </s>
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parents:
diff changeset
4068 <s>E si dispongano gli acquedotti e le fosse in guisa che formino tal’hora
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diff changeset
4069 angoli retti, tal’hora ottusi ed acuti, tal’hora più o meno ottusi ed acuti
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4070 ne’ termini de’ campi secondo la dispositione e secondo ‘l sito e secondo la
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4071 figura loro. </s>
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parents:
diff changeset
4072 <s>Oltre acciò si dee osservare che acciochè le sponde de’ fossi sieno più
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parents:
diff changeset
4073 stabili e non isgrottino e dilombino e riquoprino ‘l fondo o riempiano le
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parents:
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4074 fosse fa di bisogno che le sponde loro non sieno tirate a piombo
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parents:
diff changeset
4075 <lb></lb>//<lb></lb>e per linea perpendicolare e ad angoli retti sopra ‘l piano del
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parents:
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4076 fondo; ma per linea inclinante verso le parti del campo e declinante dal
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parents:
diff changeset
4077 vano della fossa, la quale faccia nel fondo l’angolo ottuso, come s’usa ne’
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parents:
diff changeset
4078 fossi delle fortezze. </s>
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4079 <s>Io dico l’angolo interno, non l’esterno, il quale è acuto come l’angolo ABC.
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parents:
diff changeset
4080 e BCD. è ottuso e interno, e l’angolo ABE. è acuto ed esterno, il quale si
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parents:
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4081 costituisce nella base della sponda, ed essendo acuto le linee che lo
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parents:
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4082 comprendono, cioè AB. BE. inclinandosi e ristregnendosi l’una verso l’altra,
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parents:
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4083 son cagione che ‘l detto angolo ha proprietà di semare e ritenere ‘l
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parents:
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4084 terreno. </s>
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parents:
diff changeset
4085 <s>Senza che pell’inclinamento della linea AB. verso la BE. si fa sì che la base
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parents:
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4086 del terreno della sponda è maggiore e perciò più atto a ritenere il peso
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parents:
diff changeset
4087 della terra; perciò ch’è più grande ‘l pesamento di tutta la cosa grave che
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4088 vi si ferma sopra come si vedrebbe tirando una perpendicolare dal ponto A
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4089 sopra la linea EC. E come si potrebbe mostrare per l’assioma 15 posto da
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4090 Cristofan Clavio nel primo libro d’Euclide, ritrahendolo da Proclo. </s>
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4091 <s>Perciochè, poste uguali le tre grandezze GF. ID. AB. e aggiunte due grandezze
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4092 disuguali alle GF. e alle ID., cioè EG. alla GF. e CI. alla ID. di modo che
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4093 EG. sia maggiore e CI. minore e dalla EG. si tagli la GH. uguale alla CI. di
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4094 modo che EH. sia l’avanzo in cui la grandezza GE. avanza la CI. aggionta
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4095 alla ID. e perché pel 2° assioma alla GF. e alla ID. supposte uguali, si
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4096 sono aggionte CI. ed HG. uguali, tutta la CD. sarà uguale a tutta la HF.
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4097 Adunque, poste le gionte disuguali tutta la EF. avanza tutta la CD. col
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4098 medesimo avanzo che la gionta EG. avanza la CI. Ma si è detto la AB. esser
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4099 uguale alla ID. GF. adunque tutta la EF. avanza maggiormente la AB., cioè
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4100 con l’escesso di tutta la EF. verso tutta la CD. e della gionta EG. verso
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4101 CI. preso due volte. </s>
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4102 <s>Adunque la base EF. essendo maggiore della CD. sarà molto maggiore della AB.
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4103 Ma CD. ed AB. sia ‘l peso, adunque sarà minore del posamento EF. adunque
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4104 sarà molto minore il peso AB. Oltre acciò qua quale è l’angolo nel fondo
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4105 della fossa <pb pagenum="folios 73v-74r"></pb>tale dee esser nella sponda del
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4106 campo come B. ed A. E questa maniera di tagliare ‘l terreno non a piombo ne’
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4107 cavamenti delle fosse si dice tagliare a scarpa e a sperone, o vero a
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4108 barbacane ed è assai in uso nella fortificatione poiché non si potrebbero
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4109 tenere intere le sponde de’ fossi delle fortezze se no’ si tagliassero a
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4110 scarpa, né ‘bastioni, né ‘terrapieni potrebbero già mai mantenersi in piedi
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4111 se non fussero fatti a scarpa. </s>
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4112 <s>Ma per tornare al compartimento de’ campi non si dee tralassar che in essi si
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4113 comprendano i compartimenti degli horti e de’ giardini i quali si fanno per
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4114 commodità di seminar diverse specie di semi e di piantar varie ragion di
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4115 piante e d’arboli e ordinare la distribution delle parti di qualunque
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4116 superficie di terreno con misura, con ordine, con accozzamento di più varie
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4117 figure angolari, per haver in uno stesso tempo congionta con la commodità,
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4118 la vaghezza e l’ornamento che risulta dalla varietà delle figure e dalla
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4119 distintione e diversità dell’herbe e de’ fiori e dal numero
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4120 dell’accoppiamento, dalla corrispondenza e dalla proportione e dal
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4121 collegamento del tutto insieme con le parti e delle parti in fra loro. </s>
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4122 <s>Serve oltre acciò l’uso degli angoli nell’accommodar le steccate, dette
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4123 altramente scotatoie, ne’ letti de’ fiumi per riparar i campi dal corso
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4124 dell’acque tenendole lontane; perciochè debbono collocarsi in maniera
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4125 appresso le rive de’ fiumi, che con la linea retta loro facciano due angoli,
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4126 cioè l’esterno ottuso e l’interno acuto; avvertendo sempre di opporre al
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4127 corso dell’acque l’angolo ottuso, il quale essendo contenuto da una linea
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4128 inclinante verso la parte opposta al corso del fiume e declinante dalla
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4129 parte onde viene ‘l corso, ed essendo l’angolo ottuso per sua natura non
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4130 atto a ritenere per la sua larghezza ‘l cagion dello sfuggimento dell’acque;
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4131 che se li si opponesse l’angolo acuto non <lb></lb>//<lb></lb>si cagionarebbe lo
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4132 sfuggimento, ansi perché per sua natura doverebbe dar ricetto all’acque
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4133 ristregnendosi dentro le sue linee con poco intervallo non essendo impedito
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4134 dalla debilezza della materia, non fa resistenza alcuna all’impeto
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4135 dell’acque e pertanto non le ritiene ma tosto cede alla forza loro
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4136 lassandosi disfare in tutto. </s>
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4137 <s>Che la linea inclinata la quale forma l’angolo acuto rende il riparo indabile
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4138 al resistere al corso ed all’impeto de’ fiumi. </s>
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4139 <s>Nella maniera e per la ragion medesima si fanno i pilastri de’ ponti dalla
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4140 parte che è contro al corso del fiume, che gli si fanno alcuni contra forti,
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4141 o denti i quali co’ pilastri fanno angoli ottusi per cagionar lo sfuggimento
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4142 dell’acqua da ogni banda, ma vi si forma ancho un angolo acuto in faccia il
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4143 quale stando incontra al corso dell’acque divide la forza loro, dividendo
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4144 ancho l’onde. </s>
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4145 <s>Dalla qual cosa si rende sicura e stabile qualunque fabbrica di ponti fatta
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4146 nell’acque de’ fiumi. </s>
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4147 <s>Il che si renderà più chiaro con l’esempio sensato //</s>
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4148 </p>
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4149 <p type="main">
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4150 <s></s>
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4151 </p>
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4152 </chap>
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4153 <pb pagenum="folios 74v-75r"></pb>
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4154 <chap>
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4155 <p type="head">
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4156 <s>L’uso degli angoli nel disegno</s>
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4157 </p>
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4158 <p type="head">
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4159 <s>Cap. 22</s>
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4160 </p>
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4161 <p type="main">
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4162 <s>Trovandosi l’uso degli angoli principalmente nella Geometria, sì come
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4163 abastanza da Euclide in diversi luoghi del suo volume degli “Elementi” e
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4164 come in parte è stato accennato da noi nel cap. 21 di questo lib. ed essendo
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4165 la scienza pratica del disegno subalternata alla Geometria; perciò che si
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4166 serve di ponti, di linee, di superficie e di corpi, come sensibili e
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4167 materiali e tutte cose che si dichiarano e si provano nella Geometria, come
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4168 in scienza superiore; in essa anchora apparirà manifestamente l’uso degli
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4169 angoli. </s>
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4170 <s>Che ciò sia vero rimirinsi le posture, le movenze e varij aspetti delle
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4171 figure di ciascuno animale, l’elevaioni delle piante sopra ‘l piano e delle
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4172 cose inanimate, e vedremo chiarissimamente l’uso degli angoli. </s>
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4173 <s>Perciò che la figura dell’huomo posta in piedi sopra ‘l piano forma la
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4174 perpendicolare, la quale, cadendo su piano forma angoli retti, e secondo che
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4175 si muove varia l’angolo, ed in se stessa piegandosi moltiplica gli angoli. </s>
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4176 <s>Onde segue che per disegnar tal figura nel piano sia di bisogno formar
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4177 solamente prima una perpendicolare e sopra essa poi disegnar la figura od
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4178 almeno primieramente immaginarvela, la qual linea necessariamente farà gli
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diff changeset
4179 angoli retti, sì come è manifesto per la decima def. del primo di Euclide. </s>
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diff changeset
4180 <s>Ma avanti che si venga all’intera dichiaratione di quest’uso è necessario
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diff changeset
4181 mostrar tutte le specie degli angoli che appartengono alla scienza, ed alla
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diff changeset
4182 pratica del Disegno. </s>
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diff changeset
4183 <s>Tre sono le ragioni degli angoli, delle quali si serve il disegno, ciò sono
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diff changeset
4184 rettilinei, curvilinei, misti. </s>
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diff changeset
4185 </p>
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diff changeset
4186 <p type="main">
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diff changeset
4187 <s>I rettilinei non si adoperano se non con le linee inter <lb></lb>//<lb></lb>ne le
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4188 quali son rette. </s>
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diff changeset
4189 <s>E questi sono o retti o ottusi od acuti, i quali si veggano secondo la varia
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diff changeset
4190 positione e secondo la diversa movenza della figura.</s>
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diff changeset
4191 </p>
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diff changeset
4192 <p type="main">
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diff changeset
4193 <s>I curvilinei si adoperano in fra le parti e fra ‘congiognimenti delle membra
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4194 e de’ muscoli delle figure. </s>
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diff changeset
4195 <s>E questi o sono nelle parti concave al più o si trovano nelle convesse. </s>
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4196 </p>
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diff changeset
4197 <p type="main">
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4198 <s>Finalmente gli angoli misti saranno que’ ch’im parte son formati da linee
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4199 rette e in parte da linee curve, e si trovano altresì ne’ luoghi concavi e
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4200 ne’ convessi delle membra.</s>
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4201 </p>
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diff changeset
4202 <p type="main">
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4203 <s>Dopo la consideration degli angoli è necessario considerar tre stati secondo
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4204 quali si possan trovar le figure degli animali e specialmente dell’huomo: o
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4205 come ferma e quieta, overo come movente se stessa: come ferma e quieta, sarà
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4206 o posta in ginocchioni o ‘n piedi o sedente: come movente se stessa, cioè o
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4207 correndo o saltando o camminando o muovendo braccia, mani, o piedi.</s>
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4208 </p>
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4209 <p type="main">
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4210 <s>Si è detto avanti doversi stabilire una linea retta interna e occulta che
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4211 cada perpendicolarmente sopra la linea del piano e ‘n torno ad essa
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4212 disegnarsi qualunque figura in piedi. </s>
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4213 <s>Col mezzo di questa linea la figura posta in piedi farà due angoli retti
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4214 sopra ‘l piano; né senza quest’uso di tali angoli si può regolarmente
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4215 disegnar una figura, non possendosi disegnar senza la linea perpendicolare
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4216 che si parta dal mezzo della fontanella della gola e passando pel mezzo del
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4217 collo del piede ferisca ‘l piano ad angoli retti. </s>
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4218 <s>Come si vede nella presente figura, dove la linea AB. è ‘l piano e la CD. <pb pagenum="folios 75v-76r"></pb>è la perpendicolare che forma due angoli retti
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4219 da ogni banda, cioè CDB. e ADC. disegnando la figura in faccia come ancho in
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4220 profilo. </s>
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4221 <s>Quando Aristotile nella trentesima quistione delle Meccaniche rendendo la
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4222 ragione perché mentre l’huomo postosi a sedere e levandosi in piedi faccia
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4223 con la gamba e con la coscia e col petto l’angolo acuto dice che tutto quel
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4224 che è uguale è cagion di quiete in qualunque luogo e l’angolo retto è angolo
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4225 d’egualità, pertanto esser cagione dello star fermo in piedi. </s>
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4226 <s>E quando dice angolo retto non intende l’angolo retto formato sopra ‘l
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4227 pavimento, benchè anchora così lo potesse intendere, ma quello che si forma
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4228 dalla linea perpendicolare sopra la terra; perciochè considerà l’huomo come
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4229 corpo grave il quale con la sua linea perpendicolare si muova verso la
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4230 superficie convessa e sferica della terra ad angoli simili, i quali sono
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4231 angoli retti sferali, non che si muova ad angoli retti verso ‘l pavimento;
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4232 perciochè considera la principale intensione delle cose gravi, che ricevono
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4233 la qiuete loro nel centro, o vicino al centro; che poi l’huomo
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4234 interpostosili lo impedimento del piano non possa andar a posarsi sopra la
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4235 circonferenza della terra; per accidente si ferma sopra esso ad angoli
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4236 retti, e seguendo questo come seconda intensione nel qual piano è necessario
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4237 che stia fermo ad angoli retti; perciochè qualunque l’huomo non sia nella
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4238 circonferenza o nel centro della terra, con tutto ciò la sua perpendicolare
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4239 è collo <lb></lb>//<lb></lb>cata ad angoli retti , come si può vedere in questa
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4240 figura , la dove KBL è la stessa N. il centro CD ‘l pavimento nella terra
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4241 EF. il pavimento sopra terra: e così GE: AB. la perpendicolare dell’huomo
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4242 posto in piedi; e così AM. ed HI. le quali perpendicolari formano angoli
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4243 retti nelle linee de’ piani loro, e questi son tutti uguali; perciochè
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4244 ovunque sieno son sempre uguali, come è manifesto per la decima definitione
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4245 del primo d’Euclide e pel 12° assioma. </s>
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4246 <s>Ed è natura degli angoli retti esser sempre uguali. </s>
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4247 <s>Ma nella circonferenza della terra formano angoli simili a’ retti sì come
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4248 dice Aristotile; ma in vero si appellano retti sferali, e si dice che le
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4249 linee vi cadono sopra ad angoli retti sferici. </s>
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4250 <s>Quando adunque ‘l Filosofo dà principio allo scioglimento della questione,
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4251 accenna due principij in fra loro contrarij, ciò sono:</s>
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4252 </p>
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4253 <p type="main">
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4254 <s>L’ugualità degli angoli retti in qualsivoglia cosa è cagion di quiete.</s>
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4255 </p>
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4256 <p type="main">
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parents:
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4257 <s>La mutation degli angoli retti negli acuti e negli ottusi, ed in più o meno
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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4258 acuti e ottusi è cagion di movimento.</s>
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4259 </p>
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diff changeset
4260 <p type="main">
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4261 <s>Adunque sì come gli angoli retti son cagione di riposo, il quale si fa o
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diff changeset
4262 stando in piedi (sì come si è accennato) o stando a sedere o giacendo, così
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4263 l’uso di essi può esser cagione di disegnare o dipegnere o scolpire le
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4264 figure ritte, giacenti e sedenti. </s>
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4265 <s>Ma avanti che si dimostri tale uso, la regola di far le figure sedenti,
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4266 bisogna osservar ciò che dice Arist. nella medesima questione. </s>
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4267 <s>Egli facendo comparatione dello stare in piedi con lo stare a sedere dice che
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parents:
diff changeset
4268 standosi in piedi la testa e ’l piede son collocati <pb pagenum="folios 76v-77r"></pb>nella medesima linea perpendicolare. </s>
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4269 <s>Come la linea AC posta nel piano AB in cui C. rappresenti la testa e l’A il
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4270 piede. </s>
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4271 <s>Ma sedendo il capo non è nella medesima linea che ‘l piede, ma il piede e ‘l
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4272 capo sono nelle linee parallele, come essendo DE. ed FG. perpendicolari nel
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4273 piano AB. in fra loro parallele, la testa dell’huomo sarà nel punto D. della
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diff changeset
4274 DE. e ‘l piede nel punto G. della GF., di modo che sì come gli angoli CAE. e
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diff changeset
4275 AH fanno star fermo in piedi l’huomo, son cagione che si disegni in tal
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diff changeset
4276 positura, così gli angoli DEG. DLF. il fanno posarsi a sedere e danno la
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diff changeset
4277 regola per disegnarlo regolatamente e con facilità in tal positura. </s>
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4278 <s>Proposta la linea del piano AB. perciochè si dee sempre proporre ‘l piano
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diff changeset
4279 avanti che si facciano le figure, o sieno sole, o molte insieme unite, come
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diff changeset
4280 accade ne’ componimenti delle historie, si dee tirare una perpendicolare
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diff changeset
4281 d’altezza uguale all’altezza che si mostra dall’huomo sedente e sia la linea
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diff changeset
4282 CD. cha faccia angoli retti sopra ‘l piano, di poi presa la misura della
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diff changeset
4283 coscia, cioè dal ginocchio fine all’estremità della natica, che si fa
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diff changeset
4284 replicando la misura della testa tante volte quante formano la sua
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4285 lunghezza, sì come è costume de’ pittori e degli scultori: e si tagli la
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diff changeset
4286 linea del piano secondo la misura detta; per esempio nel ponto F. Quindi nel
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diff changeset
4287 medesimo modo presa la misura della gamba e si determini nel ponto E. e da
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4288 esso si tiri la perpendicolare EF. parallela alla CD. e per la terza del
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diff changeset
4289 primo d’Euclide, della CD, si tagli una linea uguale alla EF., cioè nel
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diff changeset
4290 ponto B. di modo che GD. sia uguale alla EF. e fianalme si congionga EG. di
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diff changeset
4291 maniera che <lb></lb>//<lb></lb>vi sieno due angoli retti, detti angoli della
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diff changeset
4292 quiete, ciò sono CGE., EFB. Perciochè l’huomo sedendo fa due piegature, una
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diff changeset
4293 concava nell’anca, il quale si rappresenta dall’angolo retto CGE. e l’altra
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diff changeset
4294 convessa nel ginocchio, la quale si rappresenta per l’angolo GEF. Hora per
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4295 far la figura sopra le linee CG. GE. EF. è necessario tirare una linea dal
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4296 ponto G. al ponto E, di modo che si formi un triangolo rettangolo CGE. Si
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4297 divide poi il lato GE in due parti uguali nel ponto .I. si tiri una linea
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4298 dal ponto C. che termini nella GE. in detto ponto. </s>
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diff changeset
4299 <s>Finalmente, sopra questi lineamenti si disegni la figura ponendo ‘l ponto C.
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4300 nel mezzo della testa e ‘l ponto .I. nel collegamento della coscia e del
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4301 corpo e ‘l ponto E. nel ginocchio e ‘l ponto F. nel piede, cosa apparisce
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4302 nella figura. </s>
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4303 <s>Se adunque l’angolo retto è cagion di quiete e ci dà commodità di poter
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4304 disegnar le figure nella lor quiete, per opposito segue, che l’angolo non
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4305 retto sia cagion di movimento e dia la regola di formar le figure in atto di
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4306 mostrar qualche movenza; adunque, mutandosi positura, si muta angoli e
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4307 scambiandosi angoli si muta positura. </s>
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4308 <s>Perciò che stando l’huomo ritto forma gli angoli retti nel pian della terra,
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4309 così ancho stando a sedere forma angoli retti nel piano e nella sedia; ma
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4310 quando si leva da sedere muta angoli; perciochè ‘l rizzarsi è una maniera di
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4311 movimento.Ma per intender meglio questo movimento bisogna osservare, che se
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4312 standosi ritto ‘l capo e ‘l piede sono nella medesima linea, e sedendosi
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4313 sono <pb pagenum="folios 77v-78r"></pb>nelle linee parallele ed in amendue le
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4314 maniere si formano angoli retti nel piano della terra: e levandosi da seder
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4315 bisogna che si muti l’angolo retto in acuto; perciochè colui che siede per
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4316 levarsi da sedere dee accommodarsi in maniera, che’piedi venghino nella
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4317 medesima linea perpendicolare, né può fagli tornar nella medesima linea, se
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4318 prima non accorda i piedi in modo che venghino a dirittura del capo,
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4319 ritirandoli indietro e se non spegne ‘l capo e ‘l petto avanti; e così di
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4320 due angoli retti si fanno due acuti nel piegamento concavo e nel convesso; e
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4321 fatti tali angoli tosto l’huomo si leva in piedi. </s>
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4322 <s>Di modo che quelle linee moltiplicate nel piegamento si fanno una linea sola
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4323 nel levarsi da sedere e come nel posarsi a sedere una sola linea divien più
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4324 linee, sì come insegna Arist. nel primo cap. del lib. </s>
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4325 <s>Del movimento degli animali dicendo in questo tenore:</s>
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4326 </p>
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4327 <p type="main">
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4328 <s>E’ impossibile, che si muova alcuno animale, non havendo in sé alcuna parte
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4329 immobile.</s>
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4330 </p>
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4331 <p type="main">
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4332 <s>Perciochè nelle Meccaniche insegna che ciò che si muove sempre si dee muovere
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4333 sopra una cosa immobile. </s>
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4334 <s>Onde avviene che gli animali habbiano i piegamenti che gli servono in luogo
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4335 di centro e tutta la parte dov’è ‘l piegamento è ‘n potenza e in alto è una
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parents:
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4336 e due: è retta e piegata: e quella che si muta è in potenza avanti ‘l
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parents:
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4337 piegamento. </s>
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4338 <s>E quando alcuno animale si muove e si piega, una parte si muove ed un’altra
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4339 rimane immobile, come si può vedere con l’esperienza ed in questo
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4340 essempio.La parte B. che è nel centro è immobile, la DB. si muove verso BE.
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parents:
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4341 e la AB. verso BC. e di nuovo la EB. torna in DB. tanto che vien per dritto
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4342 BC. Onde <lb></lb>//<lb></lb>di più linee se ne fa una sola in alto dove prima era
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4343 in potenza e di più piegate si forma una diritta: così avverrebbe se fusse
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4344 la figura piegata verso G. perciochè dirizzandosi BG. si muova verso BD. e
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parents:
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4345 se ne farebbe una stessa linea. </s>
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parents:
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4346 <s>Di qui possiamo ritrarre che piegandosi l’huomo od altro animale con la parte
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4347 che si muove e con l’altra che rimane immobile si formano due linee che
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4348 fanno nel piegamento là dove è ‘l contanto commune, l’angolo retto, l’ottuso
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4349 o l’acuto, secondo la maniera del piegamento. </s>
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4350 <s>Né si può far piegatura alcuna senza l’angolo. </s>
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4351 <s>E questo è fatto conforme alla lieva , dove fa di mestiero porre una parte
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parents:
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4352 immobile per sotto lieva. </s>
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parents:
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4353 <s>Quindi adunque si ritrahe che formandosi gli angoli che fanno ‘l piegamento
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parents:
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4354 proprio della movenza del levarsi in piedi facilmente si faranno le figure
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4355 in atto di levarsi da sedere, in quella guisa che insegna Arist. nelle
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4356 “Meccaniche”, la qual cosa dimostraremo con la figura qui appresso.</s>
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4357 </p>
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4358 <p type="main">
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4359 <s>Volendo adunque tal figura levarsi da sedere, bisogna che in uno stesso tempo
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diff changeset
4360 si muova la linea AB. e venga in FB. e la CD. e venga in CE. in modo che vi
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4361 resti una linea immobile e sia CB. intorno la quale sia ‘l movimento
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diff changeset
4362 perciochè muovasi la linea AB. sopra ‘l centro immobile B. si muove secondo
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4363 la portione del cerchio AF. muovasi la CD. sopra ‘l centro immobile E. si
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4364 <pb pagenum="folio 78v - attached piece 79r"></pb>muove secondo la portion
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4365 del cerchio DE. di modo che come la AB. vien nella FB. così la CD. nella CE.
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4366 e come B. è centro immobile, così C. è centro immobile; adunque tutta la
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diff changeset
4367 linea che si è tirata da un centro all’altro, cioè CB. è immobile, adunque
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4368 come intorno a’ centri così intorno a tal linea si fanno questi movimenti. </s>
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4369 <s>Ma perché i detti centri son posti in ugual distanza, perciò le linee FB. e
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4370 CD. son parallele; oltre che son parallele anchora perché fanno gli angoli
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diff changeset
4371 alterni FBC. e BCE. uguali, sì come è manifesto per la ventinovesima del
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diff changeset
4372 primo d’Euclide. </s>
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4373 <s>E perché qui son due triangoli ABF. CGE i quali hanno le basi uguali e perciò
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4374 ancho i lati uguali, cioè ciascuno a ciascuno, come sarebbe chiaro
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diff changeset
4375 convertendo la quarta del .po. sì come si converse dal Clavio sopra lo
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diff changeset
4376 scholio dell’ottava del .po.. Adunque AB. sarà uguale a GE. ed FB. a CE. ma
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4377 CE. è uguale ad EG. adunque CE. sarà uguale ad AB. ed AB. e GE. sono una
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4378 medesima linea AE. che GB. è una parte di mezzo che le congiogne
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4379 dirittamente. </s>
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4380 <s>Adunque il termine della CE. verrà al diritto sotto ‘l ponto A nella medesima
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4381 linea. </s>
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diff changeset
4382 <s>Ma habbiamo detto nel ponto A. esser la testa e nel ponto E. ‘l piede;
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4383 adunque ‘l piede verrà sotto ‘l capo nella medesima linea muovendosi, mossa
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4384 la CD. in CE. e BE. ritornando in BA. Adunque all’hora si levarà in piedi,
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4385 quando’l ponto D verrà nel ponto E. e ‘l ponto F. tornarà nel ponto A.
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4386 cominciando ‘l movimento della figura dal movimento innanzi di AB. verso FB.
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4387 al quale succede ‘l ritiramento indietro di CD. verso BE. e questo è
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4388 movimento naturale; perciochè naturalmente sempre due membra <lb></lb>//
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4389 [Attached piece (folio 79) not transcribed.] <lb></lb><pb pagenum="attached drawing 79v - folio 80r"></pb>si muovano in parti
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4390 contrarie, mentre si muove ‘l tutto, come si vede nel caminare, che
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4391 spegnendosi avanti la gamba destra, si tira adietro la spalla e ‘l braccio
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4392 sinistro: e spegnendosi avanti la gamba sinistra si muove indietro la spalla
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4393 e ‘l braccio destro: e spegnendosi avanti la spalla diritta, si manda
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4394 indietro la gamba manca: e sporgendo innansi la spalla manca si tira
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4395 indietro la gamba dritta, come si può veder ancho nel caminar degli animali
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4396 di quattro piedi e si può ritrar da Arist. nel lib. </s>
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4397 <s>Del caminar degli animali, che negli huomini le spalle e le braccia e le
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4398 gambe fanno quello che le gambe davanti, con quelle di dietro degli animali
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4399 di quattro piedi. </s>
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4400 <s>La forza di questo movimento non è altro che la forza della lieva; perciochè
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4401 la linea FB. è ‘l movente e la GE. e CB. la lieva, e la DE. il sottolieva;
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4402 che mentre FB. si muove innanzi e CE. indietro premendo ‘l piano DE. CB. si
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4403 solleva e inalza tutto ‘l peso. </s>
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4404 <s>Tutti questi piegamenti di linee e tutti questi angoli che son cagion di
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4405 questa movenza ci danno la regola con la quale agevolmente possiamo formare
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4406 una figura humana in atto di levarsi da sedere, sì come si vede chiaro
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4407 osservando la figura e tirando i suoi contorni intorno alle linee che fanno
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4408 gli angoli acuti FBC. BCE. Ma perché havemo anchor accennato che giacendosi
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4409 si formano gli angoli retti; perciochè in tal positura si sta in riposo e ‘n
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4410 quiete, perciò si dee mostrar come la figura humana giacente faccia angoli
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4411 retti sopra la superficie della terra. </s>
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4412 <s>E’ cosa chiara che tal figura dimostra un corpo che ha tutte le sue misure in
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4413 fra le quali la misura della grossezza forma la linea perpendicolare in più
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4414 parti, la quale cade nel <pb pagenum="folios 80v-81r"></pb>piano ad angoli
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4415 retti, come per essempio nel piano AB. giacendo la figura D. con la testa
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4416 con l’anche, con le ginocchia e co’ piedi fa tante linee perpendicolari, le
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4417 quali stanno ad angoli retti sopra ‘l piano AB. e se si havesse a muovere
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4418 bisognarebbe che la linea FH venisse nella perpendicolare GH. che così
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4419 sarebbe sedente nel piano; e volendo rizzarsi bisognarebbe che la GH. si
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4420 muovesse verso IL. e la IL. verso la MN. e la MN. e la QR. verso OP. mutando
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4421 sempre gli angoli retti in acuti, sì come già s’è detto e secondo i
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4422 lineamenti del giacere e del muoversi a sedere e del rizzarsi; e secondo la
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4423 forza degli angoli contenuti sotto tante linee fra loro variamente piegate
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4424 si potrà con facilità disegnar le figure in tutte le muovenze e ‘n tutte le
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4425 positure, sempre procedendo dalla figura della quiete a quella del
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4426 movimento, da quella del movimento a quella della quieta e quindi un’altra
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4427 volta a quella della quiete, cioè dal giacere al rizzarsi a sedere; dal
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4428 rizzarsi al sedere, dal sedere al levarsi in piedi, dal levarsi in piedi
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4429 allo star ritto. </s>
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4430 <s>Ma avanti che poniamo fine a questo discorso, porremo alcune figure di
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4431 movenze insieme co’ lineamenti e con le regole e colli angoli ne’ quali
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4432 consistono le diversità delle movenze, dalle quali agevolmente si potrà
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4433 imparare a mettere insieme le figure.</s>
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4434 </p>
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4435 <p type="main">
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4436 <s>La muovenza di questa figura si fa col mezzo di due triangoli almeno uno
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4437 acut’angolo, cioè ACE <lb></lb>//<lb></lb>e l’altro ottus’angolo, cioè ECD,
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4438 aggiontivi due altri triangoli, cioè uno scaleno DFE. ed uno equicure GDF.
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4439 per l’accomodamento delle braccia. </s>
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4440 <s>Dove nel muover il passo AE. si fa la base del triangolo acutangolo e nello
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4441 spegnere ‘n fuore la testa col petto si forma ‘l triangolo ottusangolo la
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4442 cui base è la perpendicolare ED. perciochè acciochè si faccia che la figura
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4443 non sia cadente, bisogna che ‘l capo sia appiombo sopra ‘l piede.</s>
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4444 </p>
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4445 <p type="main">
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4446 <s>Questa figura forma una movenza che si può disegnare col mezzo di tre
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diff changeset
4447 triangoli, due rettangoli AFD. FCG. ed uno ottuso angolo DEF. Il passo AD. è
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4448 la base del triangolo ADF. la base dell’ottusangolo è FD. nella quale si
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4449 regge l’angolo del piegamento E. la qual base è per dritto del lato FC. del
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diff changeset
4450 triangolo FGC. nel cui angolo retto è ‘l sito della testa e nell’angolo
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diff changeset
4451 acuto appresso alla base, cioè cioè nel CGF. el termine dello stendimento
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4452 del braccio.<pb pagenum="folios 81v-82r and attached piece"></pb></s>
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4453 </p>
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4454 <p type="main">
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4455 <s>Questa movenza non solamente si serve di triangoli rettangoli, acutangoli,
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4456 ottusangoli, ancho d’un parallelogrammo. </s>
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4457 <s>Perciò che vi sono sei rettangoli, ciò sono FCD. DEF. FCH. DHG. DHI. CHI. vi
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4458 è uno acutangolo CID. uno ottusangolo CFH. e due acutangoli, uno equilatero
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4459 CDI. e uno equicruro NPE ed un parallelogrammo CDEF. Perciochè la distanza
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4460 dal luogodel ginocchio D. alla mano E. si fa dalla longhezza del
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diff changeset
4461 parallelogrammo, così la distanza dell’anca al collo o al mento od alla
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diff changeset
4462 spalla o l’altezza che dee porsi fra’l ginocchio al termine dell’anca e così
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4463 fra la palma della mano all’ultimo della spalla si costituisce dall’altezza
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4464 del parallelogrammo: il luogo della testa si pone fra due angoli del
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4465 triangolo CFH. Il quale si forma tirata la linea L sopra gli angoli DF. ed
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4466 allongata fine ad N quanto è la meta della CD. Il luogo dell’anche e delle
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4467 natiche sarà nell’angolo C. lo ingrossamento del corpo sarà sopra la base
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4468 del triangolo HCF. e le spalle e reni G sopra ‘l lato CN del triangolo CNF.
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4469 e divisa la linea FD in due parti uguali nel ponto G. si tiri da esso, sopra
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4470 ‘l ponto H una linea tanto longa che la G. vada al ponto I. Di poi tirata
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4471 dal ponto D. una linea retta che passi pel ponto L. infine al ponto K. il
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4472 quale sia parallelo col ponto C. di modo che da I. a K. sia una linea uguale
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4473 alla DC. e tirata una linea dal .L. al C. che formi un triangolo rettangolo
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4474 nella linea CI. che sia la <lb></lb>//<lb></lb>metà d’uno triangolo equilatero CDI.
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4475 Nella qual linea sarà la coscia manca e nel punto I il ginocchio e nella
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4476 linea KI la gamba e ‘l piede e nella rimanente DL uguale alla IK. la gamba e
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4477 ‘l piè destro: e nella CD la coscia dritta. </s>
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4478 <s>Di modo che CDL. fanno ‘l piegamento destro della gamba con la coscia: e CIK
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4479 il piegamento sinistro: ed FCD ‘l piegamento del corpo con le cosce e con le
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4480 gambe e lati del triangolo EUP. con la base EU allongata verso M. danno i
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4481 luoghi alle braccia.</s>
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4482 </p>
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4483 <p type="main">
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4484 <s>Questa figura benchè sopra ‘l piano faccia gli angoli retti e di quiete, con
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4485 tutto ciò non è rappresentativa di quiete ma d’atto di riverenza e
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4486 d’adoratione; perciochè gli stati di quiete non sono più di tre giacendo,
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4487 sedendo e stando ritto. </s>
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4488 <s>Si disegna sopra la superficie del piano CD, tirata la perpendicolare AB e la
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4489 parallela GD. e la FG. ed allontanandosi dal ponto B. quanto è naturalmente
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4490 longa una gamba si formi un triangolo rettangolo nella CD. facendo la base
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4491 EB. el maggior lato opposto al maggiore angolo B. e ‘l lato mezzano la
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4492 perpendicolare FB. che è parte della AB. e trovata la longhezza dal
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4493 ginocchio all’estremo della natica dal ponto F. si tiri una linea di tal
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4494 longhezza cioè FG. e dalla CD. si tagli una parte uguale cioè BD. Finalmente
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4495 dal ponto G si tiri una perpendicolare al ponto D. formando un
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4496 parallelogrammo di modo che nella perpendicolare AB. si ponga la testa,
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4497 l’anca o ‘l ginocchio: nelle linee che fanno l’angolo retto FGD opposto
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4498 all’FBD si accomodi la coscia e la gamba manca: e ne’ due lati del triangolo
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4499 FB. BE. si disegni la coscia, e la gamba dritta.</s>
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4500 </p>
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4501 <p type="main">
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4502 <s>Alla precedente figura appartiene un’altra, con la quale si mostra quella
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4503 movenza che si forma nel porsi <pb pagenum="folios 82v-83r"></pb>in ginocchioni
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4504 pertanto la possono qui appresso insieme co’ lineamenti suoi, acciò si veda
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4505 la regola per disegnarla e l’uso degli angoli.</s>
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4506 </p>
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4507 <p type="main">
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4508 <s>La figura di questa movenza dimostra l’attitudine di porsi in ginocchioni, e
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4509 si fa posta la linea del piano AB. e sopra essa tirata la perpendicolare CD.
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4510 da cui allontanandosi quanto è l’altezza della sesta si faccia ‘l segno F.
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4511 sopra cui si tiri un’altra perpendicolare EF. e divisa la perpendicolare EF.
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4512 in sette parti uguali, presene quattro si formi ‘l parallelogrammo DFHI. il
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4513 quale si divida per mezzo con la linea infinita LN. quindi presa la
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4514 longhezza DF. e replicata nella linea AB., cioè nel ponto G. dal quale si
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4515 tiri una linea che passi sopra ‘l ponto L. e termini nel ponto H. e preso ‘l
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4516 compasso con esso si pigli la longhezza GD. e si trasporti in GO. tirando
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4517 l’arcuatione: e presa la longhezza LG., fatto centro L. si faccia un’altra
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4518 arcuatione che tagli la prima nel segno O. dal quale si tirino le linee OL.
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4519 OG. onde si formi un triangolo acutangolo la cui base divisa in tre parti
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4520 uguali dal taglio della terza, cioè dal P. si tiri la linea PL. finalmente
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4521 sopra la linea IH. si costituisca un triangolo rettangolo tirata la linea
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4522 IE. e presa una delle .7. portioni con essa si tagli la linea MN. nel ponto
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4523 N. dal quale a ponti FH. si tirino due linee rette, le quali chiudino un
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4524 triango <lb></lb>//<lb></lb>lo ottusangolo HNF. la cui base nella perpendicolare EF.
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4525 ed è FH. La retta si disegnarà nell’angolo E. facendo che l’estremo della
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4526 perpendicolare sia nell’orecchio, la spalla, il corpo e parte della natica e
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4527 dell’anca si farà dentro ‘l triangolo IEH. la coscia e la gamba dritta nella
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4528 linea nelle linee HL. LP. e nelle linee HN. NF. la coscia e la gamba manca. </s>
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4529 <s>Il calcagno del piè destro sarà nel ponto P. e la ponta del piede nel ponto
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4530 A. e ‘l ginocchio nel ponto L.: il principio della coscia sinistra sarà nel
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4531 ponto H. il ginocchio nel ponto N., la pianta del piede nel ponto F. Il
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4532 braccio cominciarà dal terzo della linea EI. presso al ponto E. Nella qual
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4533 figura si vede che tutti i piegamenti delle parti si fanno negli angoli, i
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diff changeset
4534 quali parte sono ottusi, e parte acuti. </s>
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4535 <s>Ma perché con l’essempio di queste poche movenze ciascun diligente
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diff changeset
4536 disegnatore , o pittore ,o scultore potrà facilissimamente disegnar
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diff changeset
4537 qualunque figura di qualsivoglia animale secondo qualunque maniera di
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diff changeset
4538 movenza, di gesto e di positura; perciò non pigliarò cura di porvi davanti
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diff changeset
4539 altri essempi. </s>
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4540 <s>Ma solamente aggiognerò questa consideratione degna d’esser da ciascuno
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diff changeset
4541 osservata che a tutte queste regole si dee anteppor la cognition della
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diff changeset
4542 misura dell’altezza della figura humana e delle sue membra, acciò che con
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diff changeset
4543 essa possiamo determinare l’altezza delle perpendicolari le grossezze e
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diff changeset
4544 longhezze de’ corpi e delle braccia, delle cosce e delle gambe e de’ piedi. </s>
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diff changeset
4545 <s>Pertanto acciò che’l nostro discorso non sia manchevole, ed acciò altri non
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diff changeset
4546 si habbia a dolere di noi, quando non insegnassemo tal misura, ne faremo un
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diff changeset
4547 racconto nell’ultimo di questo capitolo.</s>
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diff changeset
4548 </p>
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diff changeset
4549 <p type="main">
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4550 <s>La statura della figura humana, come si ritrahe da Vitruvio, nel cap. primo
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4551 del 3° libro, può ricever tre maniere di misure secondo tre ragioni di
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diff changeset
4552 misurare. </s>
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diff changeset
4553 <s>Perciò che <pb pagenum="folios 83v-84r"></pb>la prima si prende dalla testa,
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4554 cominciandosi dalla estremità del mento e terminando nelle radici dei
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diff changeset
4555 capelli; la seconda si piglia dal mento alla sommità della testa; la terza
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diff changeset
4556 dalla sommità del petto alle barbe de’ capelli.</s>
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diff changeset
4557 </p>
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diff changeset
4558 <p type="main">
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4559 <s>Se si prende nel terzo modo, si dividarà tutta l’altezza della figura humana
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4560 in sei parti uguali e ciascuna di esse sarà un piede antico formato con
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4561 quattro palmi, ciascuno de’ quali è quattro dita, il quale è alquanto
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4562 maggiore del nostro mezzo braccio.</s>
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4563 </p>
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4564 <p type="main">
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4565 <s>Se nel secondo compartiremo l’intera altezza della statura dell’huomo in otto
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4566 parti, che saranno otto teste, e ciascuna testa corrisponderà a nove once
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4567 del nostro braccio e tre quarti d’oncia, od al più ad un terzo.</s>
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4568 </p>
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4569 <p type="main">
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4570 <s>Se nel primo partiremo tutta l’altezza sua in dieci parti uguali, cioè in
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4571 dieci teste, e ciascuna si dimostra dallo spiegamento della mano,
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4572 cominciando dall’attaccatura, e terminando nella estremità del dito di
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4573 mezzo: e corrisponde a sette once del nostro braccio e un quarto d’oncia, ed
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4574 al più ad un terzo.</s>
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4575 </p>
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4576 <p type="main">
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4577 <s>Ma si avverta oltre acciò, che secondo altri si può divider tutta l’altezza
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4578 dell’huomo in quattro parti uguali, cioè in quattro cubiti, di modo che ella
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4579 arrivi all’altezza di dodici piedi, la quale, al modo nostro, si riduce a
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4580 tre braccia o poco più.</s>
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4581 </p>
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4582 <p type="main">
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4583 <s>Pomponio Gaurico, nel libro Della scoltura dell’huomo e Girolamo Cardano, De
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4584 subtilitate rerum, dividono l’altezza dell’huomo in nove teste, cioè presa
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4585 l’altezza della testa dalla estremità del mento fine ad un capello della
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4586 cima del capo: e ciò osservano negli huomini perfetti; che l’altezza de’
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4587 fanciulli è solamente quattro facce. </s>
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4588 <s>Vitruvio nello stesso luogo citato dice; il petto esser la quarta parte di
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4589 tutto ‘l corpo humano; ma non esplica in che maniera si faccia tal misura,
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4590 cioè donde prenda principio e dove termini. </s>
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4591 <s>Solamente egli dice, quasi <lb></lb>//<lb></lb>per dichiaration di questo che ‘l
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4592 cubito sia la quarta parte, ma non applica tal misura alla larghezza né alla
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4593 longhezza del petto. </s>
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4594 <s>Ma io tengo per certo il petto cominciar dal fine delle costole e terminar
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4595 nella fontanella della gola, là dove concorrono le clavicole; perciochè
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4596 tanto si estende il petto, quanto spatio si contiene sopra ‘l diaframma e
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4597 fia l’intervallo dal termine delle costole alla fontanella della gola è la
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4598 quarta parte dell’altezza dell’huomo. </s>
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4599 <s>Benchè Guglielmo Filandro non creda che ‘l petto sia la quarta parte;
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4600 perciochè dice non esser la quarta parte, ma un poco meno della quinta. </s>
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4601 <s>Ma per salvare l’oppinion di Vitruvio soggiogne che quando egli dice il
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4602 cubito esser la quarta parte la prende non secondo ‘l costume quasi di tutti
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4603 gli autori, dal congiognimento del braccio, cioè dal gombito fine al corpo o
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4604 bracciale, cioè dal congiognimento del braccio con la mano, che altramente
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4605 si dice collo della mano; ma fine all’estremo del dito di mezzo. </s>
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4606 <s>Vi sono altre maniere di misure, le quali si esplicano molto bene dal
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4607 Filandro nell’ Annotatione sopra Vitruvio, nel cap. primo del 3° libro. </s>
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4608 <s>Solamente aggiognerò per beneficio delle regole proposte doversi haver
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4609 cognitione delle misure delle braccia, delle cosce e delle gambe per poter
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4610 meglio ed a misura proportionatamente dispor i lineamenti co’ quali si
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4611 compongano e si mettano insieme le parti delle figure.</s>
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4612 </p>
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4613 <p type="main">
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4614 <s>Il braccio cominciando dalla sua congiuntura infine al collo della mano è due
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4615 teste, presa la misura nel primo modo e compresavi la mano secondo ‘l suo
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4616 maggiore stendimento sarà tre teste. </s>
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4617 <s>La coscia parimente è due teste, cioè dall’attaccamento, nell’ anguinaia
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4618 infine al congiognimento del ginocchio che è nel mezzo di esso. </s>
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4619 <s>La gamba cominciando da questo collegamento, e seguendo infine al mezzo delle
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4620 <pb pagenum="folios 84v-85r"></pb>cappolle o talloni;altresì è due teste, ma
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4621 compresa tutta la longhezza del piede è tre teste e considerata l’altezza di
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4622 tutto ‘l piede, cioè dalla pianta al mezzo delle cappolle è due teste e
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4623 mezzo. </s>
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4624 <s>Ma perché l’huomo, come perfettissimo di tutti gli animali, è misura di
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4625 tutti, sì come si ritrahe da Vitruvio e come si confermarebbe da’ Filosofi,
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4626 che vogliono che in ogni università cose, se ne ritrovi una, la quale
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4627 essendo perfettissima sia misura di tutte l’altre, pertanto non è maraviglia
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4628 che dalle regole della sua figura, che consistono nell’uso degli angoli, si
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4629 possino ritrarre le regole da formare le figure degli altri animali,
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4630 servendosi altresì degli angoli, come nel disegnar le figure humane secondo
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4631 varie positure, e secondo diverse movenze. </s>
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4632 <s>Le quali, da qualunque giuditiosissimo disegnatore osservase potranno esser
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4633 cagione del ritrovamento de’ lineamenti e degli angoli, sopra quali, con
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4634 agevolezza e con giustissima regola ciascuno potrà disegnar qualsivoglia
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4635 figura d’animale. </s>
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4636 <s>Onde non prenderemo cura di mostarne essempio alcuno, rimettendoci in tutto
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4637 alla diligenza di quantunque ingegnoso ed eccellente disegnatore.</s>
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4638 </p>
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4639 <p type="main">
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4640 <s></s>
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4641 </p>
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4642 </chap>
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4643 <chap>
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4644 <p type="head">
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4645 <s>Dell’uso degli angoli nell’arti fabrili</s>
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4646 </p>
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4647 <p type="head">
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4648 <s>Cap. 23</s>
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4649 </p>
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4650 <p type="main">
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4651 <s>Perciochè le cose che nell’uso humano sono più frequenti, sono molto meno
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4652 osservate e conosciute; onde bene spesso si vede che l’huomo non osserva né
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4653 avvertisce le cose che seco accompagnate sono e sue proprie e che non
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4654 solamente nel corpo ma nell’animo anchora si truovano per la qual cosa ne
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4655 segue una ignoranza delle cose a lui più fami<lb></lb>//<lb></lb>liari; pertanto la
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4656 ragione, la natura, e la cagion della forza e proprietà di molti stromenti
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4657 communissimamente adoprati nell’arti fabrili è al tutto ignota, non per
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4658 altra cagione se non perché nella frequenza dell’uso si nasconde il difetto
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4659 dell’osservanza e della consideratione della proprietà e delle forze loro e
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4660 quindi sorge l’ignoranza dell’origine delle loro operationi e della
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4661 dipendenza e dell’utilità che prendono dalle Matematiche e specialmente
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4662 dalla Geometria. </s>
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4663 <s>Per levare adunque la detta ignoranza, formeremo alcune considerationi sopra
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4664 alquanti stromenti dell’arti fabrili e ne trarremo la cognition
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4665 dell’utilità, che ricevono dalla Geometria, cioè dall’uso degli angoli. </s>
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4666 <s>Perciochè gli stromenti fabrili, o tagliano o forano o segano o incidono o
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4667 radono o limano o pianano o puliscono o rompono o premono o spengano o
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4668 cacciano o scagliano o scalzano ne possono mandare ad effetto dette
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4669 operationi se non col mezzo degli angoli acuti, o retti, o misti. </s>
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4670 <s>Oltre acciò gli angoli servono ancho per collegamento e così serve l’angolo
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4671 retto. </s>
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4672 <s>I quali stromenti apparte si consideraranno, riguardando minutamente
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4673 qualunque sia di loro. </s>
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4674 <s>E quindi vedremo di quanta utilità sia la Geometria verso l’arti fabrili: che
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4675 chi negasse tale utilità levarebbe uno degli usi principali delle
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4676 matematiche. </s>
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4677 <s>Ma avanti che esaminiamo tutti gli stromenti dell’arti fabrili bisogna notare
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4678 che mentre ragioniamo degli angoli si debba intendere degli angoli solidi,
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4679 non de’ piani perciochè tali stromenti son corpi solidi ma irregolari. </s>
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4680 <s>Gli stromenti che servono per tagliare o si adattano a tagliar la pietra o ‘l
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4681 legno; que’ che servono a tagliare il legno son questi: l’accetta, l’ascia,
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4682 lo scarpello, la pialla e ‘l coltello. </s>
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4683 <s>Il coltello o con due manichi che si chiama coltello a petto o con un manico,
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4684 che oltre al tagliare il legno serve ancho a diversi usi e a varie arte. </s>
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4685 <s>La pialla non è differente se non secondo <pb pagenum="folios 85v-86r"></pb>la
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4686 grandezza e secondo la figura e secondo la varietà dell’uso, che altra serve
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4687 per far piane ed uguali le tavole, altra serve per far le cornici e questa
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4688 ragion di pialla che comunemente si appella pialluzzo ha tante differenze
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4689 quante sono le maniere delle membra di qualsivoglia cornice secondo
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4690 qualunque de’ cinque ordini dell’architettura ed oltre all’havere l’angolo
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4691 solito nel taglio hanno ancho talhora a’ triangoli concavi necessarij alla
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4692 formatica degli angoli convessi delle membra che da esse si debbano formare. </s>
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4693 <s>Gli stromenti che servono a tagliare la pietra sono gli scarpelli che sono o
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4694 acuti o piani o dentati o spartiti. </s>
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4695 <s>Lo scarpello da legno overo è piano od acuto ed aognato overo incurvato, come
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4696 sono le sgobbie: e questi o si adoperano a lavorar di quadro o d’intaglio
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4697 overo a tornire. </s>
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4698 <s>L’ascia altramente detta mana non ha altre differenze che quelle della
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4699 grandezza e dell’uso; che altre asce servono per abbozzare le tavole e ‘
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4700 regoli ed altre servono per incavare, le quali più longhe, ma di taglio più
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4701 stretto dell’altre, le quali si adoperano ad incavar travi e a fare ombuti. </s>
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4702 <s>L’accetta, chiamata scure, altre si nota altre differenze che della
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4703 grandezza; ciascuno di questi stromenti riceve l’essere e l’operation sua
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4704 della natura degli angoli secondo ‘quali è formato. </s>
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4705 <s>Non senza ragione Aristotile ne’ libri dell’ Anima dice per essempio che la
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4706 scure non è veramente scure se non ha ‘l taglio, ma è scure equivocamente, e
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4707 ‘l taglio consiste nell’angolo il quale è acuto, che quanto più è acuto,
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4708 tanto più ha forza di tagliare perciochè più agevolmente entra nel legno,
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4709 penetrando con forza di lieva e a guisa di conio, ad entrar da un ponto
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4710 indivisibile, acquistando ogn’hora più di legno ed ogn’hora più ne leva. </s>
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4711 <s>Onde vediamo espressamente che ‘l taglio è collocato nell’angolo acuto, e
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4712 quando l’angolo è acuminato dalla pietra d’arrotare, allhora più taglia e
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4713 con più agevolezza ricide ogni legno, benchè sia duro, ed ogni
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4714 <lb></lb>//<lb></lb>pietra benchè dura. </s>
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4715 <s>Il medesimo accade all’ascia e ad ogni maniera di scarpello, di pialla, o di
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diff changeset
4716 coltello. </s>
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diff changeset
4717 <s>Gli stromenti che servono per forare o la pietra, o ‘l metallo, o l’osso, o
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parents:
diff changeset
4718 ‘l legno sono il trapano o ‘l succhiello, detto in altra maniera trivello. </s>
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parents:
diff changeset
4719 <s>Il succhiello non ha altre differenze che quelle le quali si prendano dalla
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parents:
diff changeset
4720 grandezza; che altri sono succhielli grossi, altri minuti, altri mezzani,
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parents:
diff changeset
4721 secondo la grandezza de’ chiodi, delle bullette, de’ cavicchi, o de perni,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4722 che si adoperano per collegamento del legname, o per altro fine. </s>
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parents:
diff changeset
4723 <s>L’operation di questo stromento consiste nel passar il legno rodendo e
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parents:
diff changeset
4724 bucando acciochè meglio faccia l’effetto suo ha bisogno di due maniere
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parents:
diff changeset
4725 d’angoli; uno è acuto, ma alquanto curvo, come è quello della ponta onde
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parents:
diff changeset
4726 comincia formarsi la vite, secondo la quale si gira, acciochè si ficchi
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parents:
diff changeset
4727 dentro ‘l legno, ed è parte convesso e parte concavo e l’altro altresì è
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parents:
diff changeset
4728 acuto, su per lato e su per la vita, ed è parte convesso e parte concavo,
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parents:
diff changeset
4729 acciochè mentre si volta tagli e roda il legno, e riceva la tagliatura nella
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parents:
diff changeset
4730 concavità della vite. </s>
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parents:
diff changeset
4731 <s>Si trova ancho un’altra maniera di succhielli, la quale si adopra
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parents:
diff changeset
4732 nell’agricoltura per far li buchi nel terreno sodo, per mettervi le colonne
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parents:
diff changeset
4733 da pergole. </s>
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parents:
diff changeset
4734 <s>Il trapano è di varie maniere; perciochè altro è ‘l trapano da legno, altro
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parents:
diff changeset
4735 da pietra, altro da metallo. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4736 <s>Il trapano da metallo si fa a ponta di diamnte ad angolo retto o ad angolo
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4737 acuto e quadrato o triangolare; il trapano da pietra si fa con angolo acuto
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4738 nel taglio, ma ‘l taglio è curvo, di modo che quando comincia a rodere la
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parents:
diff changeset
4739 pietra, muovendosi di moto circolare reflesso e ripiegato comincia in un
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parents:
diff changeset
4740 ponto; quello da legno o da osso è formato con due angoli simili ma
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4741 acutissimi, uno de’ quali serve per centro, in fra ‘ quali è ‘l taglio a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
4742 modo di lunetta. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4743 <s>Gli stromenti che adoperano a segar o la pietra, o ‘l legno, o l’osso. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4744 <s>Quelli che servono a segare il legno, o l’osso sono tutte le ragioni di
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4745 seghe, cioè grandi per segar i modelli per far le tavole le quali sono
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parents:
diff changeset
4746 grosse e lunghe, e queste sono di due maniere; percio<pb pagenum="folios 86v-87r"></pb>chè altre hanno i denti più spessi ed angoli
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4747 più acuti ed altre gli hanno radi, e d’ angoli meno acuti; quelle che
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4748 s’usano a segar le pietre sono senza denti, dritte e fatte con due angoli
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4749 retti nel filo che sega e nella grossezza quasi a modo di riga. </s>
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parents:
diff changeset
4750 <s>Queste sono di due maniere, secondo due ragion di materie da segarsi;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4751 perciochè altre sono le seghe da segar le pietre tenere, altre quelle da
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4752 segar le dure. </s>
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parents:
diff changeset
4753 <s>Le seghe che si adoperano a segar le pietre dure, e specialmente le gioie,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4754 sono di rame, le quali per loro stesse non hanno forza di segar tali pietre,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4755 ma insieme con la polvere di smeriglio, sì come ‘l solo smeriglio non può
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parents:
diff changeset
4756 segarle , ma amendue insieme, perciò che ‘l rame essendo dolce cede allo
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parents:
diff changeset
4757 smeriglio e muovendosi il conduce da qua e là su pper la pietra, tanto che
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parents:
diff changeset
4758 l’uno e l’altro entra e rode e sega le pietre benchè durissime. </s>
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parents:
diff changeset
4759 <s>Quelle che si adoperano a segar le pietre tenere sono di ferro, le quali
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4760 parimente per loro stesse non segano, ma insieme con la rena di fiume, per
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4761 la medesima ragione che segano quelle di rame. </s>
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parents:
diff changeset
4762 <s>Gli stromenti che si adattano a incidere sono i bulini, che si distinguono in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4763 due maniere, perciochè altri servono a intagliar le lamine di rame per le
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4764 stampe de’ disegni, altri si usano dagli orefici per intagliar l’oro o
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parents:
diff changeset
4765 l’argento, in qualunque maniera di lavoro. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4766 <s>E questi sono angolari, o ad angolo più o meno acuto o ad angolo retto. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4767 <s>Per radere o raschiare servono i rasoi o coltelli, i quali son fatti
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4768 acutissimi. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4769 <s>Per limare, o raspare, o pulire il metallo, il legno, la pietra e l’osso,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4770 servono le lime, che sono angolari, cioè o con angoli retti o con angoli
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4771 acuti rettilinei, o con angoli acuti curvilinei, o con angoli misti, ma sono
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4772 dritte o ritorte secondo che si richiede nella materia che si ha da polire e
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4773 da spianare, come con le lime e le raspe che si adoperano per
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4774 ri<lb></lb>//<lb></lb>metter le figure che si fanno di gitto o quelle che si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4775 scolpiscano in marmo o in altra pietra tenera. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
4776 <s>Oltreacciò le lime e le raspe, rodendo, spianano e puliscono la materia col
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4777 mezzo della multiplication degli angoli minutissimi che sono ne’ denti loro
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4778 fatti nelle superficie. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4779 <s>Gli stromenti che si adoperano per rompere la pietra o ‘l legno sono i conj o
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4780 le zeppe, le quali son fatte o di ferro o di legno, i quali acciochè rompino
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4781 e spacchino le materie con forza di lieva, bisogna che habbiano l’angolo
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4782 acuto. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
4783 <s>Quelli che premono e spengono e cacciano i chiodi, conficcandosi son fatti ad
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4784 angoli retti a squadra ed alcune volte ancho non a squadra, ma a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4785 sottosquadra, cioè ad angoli acuti appresso alla superficie piana. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4786 <s>Tali sono i martelli, i magli, le mazze, che servono ancho per spianare e
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4787 tirar ed agguagliare il ferro o ‘l rame; i modelli de’ torchij e delle viti
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4788 degli strettoi e parimente i ciselli che adoperano gli orefici per iscolpire
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4789 con bassirilievi ed opra che gli antichi Romani si appellava celata. </s>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4790 <s>Gli strumenti co’ quali si scaglia, cioè co’ quali si tagliano le pietre
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4791 levandone a scaglie a scaglie sono le subbie, cioè una ragion di scarpelli
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4792 fatti non con taglio, ma con ponta quadrata e con angolo acuto nella ponta. </s>
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parents:
diff changeset
4793 <s>Finalmente gli stromenti co’ quali si scalza e si solleva le pietre, la
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4794 terra, le muraglie, scalzando e smurando i mattoni e le pietre, sono gli
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4795 zapponi, i picconi, i pali di ferro, i martelli e le zappe. </s>
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parents:
diff changeset
4796 <s>Ma le zappe e gli zapponi si adoprano ancho nell’agricoltura per fare
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4797 scassati e per far fosse e forme; e nell’arte del fabbricare servono per
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parents:
diff changeset
4798 cavar fondamenti, pozzi, citerne e cantine; <pb pagenum="folios 87v-88r"></pb>ed
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parents:
diff changeset
4799 oltre acciò ancho la vanga e la pala serve a’ medesimi usi. </s>
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parents:
diff changeset
4800 <s>I picconi, i pali di ferro e ‘ martelli si adoperano per tagliare le
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
4801 muraglie, per ispianarle, per rompere e cavar le pietre e tagliarle, facendo
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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4802 fondamenti, incavando la terra per far cantine, pozzi, cisterne, mine,
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4803 acquedotti, o bottini, o cavando le miniere, facendo ogni maniera di
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4804 cavamento. </s>
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4805 <s>E tutti questi stromenti si adattano a tutte queste operationi con l’aiuto
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4806 dell’angolo acuto che si trova o nella ponta o nel taglio loro. </s>
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4807 <s>Perciochè gli zapponi, le zappe e le vanghe hanno l’angolo acuto ma non
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4808 quadrato nella ponta, ma taglio dritto e piano, come ancho i martelli da
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4809 muratori, nel tagilo della penna; i picconi e ‘ pali di ferro hanno l’angolo
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4810 acuto e quadrato nella ponta, a guisa di subbia, ma i picconi hanno ancho da
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4811 un’altra banda un angolo acuto non quadrato, ma triangolare non in ponta ma
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4812 in linea piana, col quale si suol far la lieva alle pietre scalzate per
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4813 muoverle dal luogo od a sollevare altro peso. </s>
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4814 <s>Oltreacciò, nel dar forza al movimento delle macchine, come degli argani,
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4815 delle viti o d’altro, servono gli angoli retti, che vi si formano i luoghi,
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4816 dove si debbano por le stanghe, o manovelle, o lieve per muover gli argani e
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4817 questi si fanno ad angoli retti. </s>
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4818 <s>Ma nelle viti che hanno la testa quadrata servono ancho gli angoli retti
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4819 perciochè ponendo la testa loro in una chiave quadrata perfettamente in modo
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4820 che la testa della vite molto bene s’incastri nel vuoto della chiave; sì che
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4821 gli angoli convessi dell’una occupino interamente gli angoli concavi
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4822 dell’altra; onde muovendosi la chiave per la resistenza e contra
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4823 <lb></lb>//<lb></lb>lieva degli angoli si fa muover la vite e li argani. </s>
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4824 <s>Per la medesima ragione e per la medesima forza si fanno girar le viti che
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4825 nel mezzo della testa loro hanno un taglio fatto a linea retta , la quale
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4826 colli estremi di essa, e col suo diametro fa angoli retti, i quali danno
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4827 vigore al movimento. </s>
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4828 <s>E così altri stromenti ed altre macchine fabrili si potrebbono mostrare, i
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4829 quali si servano del benefitio degli angoli, ma per non far maggior volume,
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4830 si tralassano, dando luogo a qualunque ingegnoso ne volesse far gionta a
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4831 questo mio trattato, e così facendo fine sigillaremo con quest’ultime righe
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4832 l’opra nostra a gloria e ad honore della infinita Sapienza e Providenza,
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4833 onde procede tutto l’ingegno e tutte le nostre questioni.</s>
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4834 </p>
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4835 </chap>
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4836 </body>
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4837 </text>
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4838 </archimedes>