Mercurial > hg > mpdl-xml-content
annotate texts/XML/echo/en/Harriot_Add_MS_6784_XT0KZ8QC.xml @ 6:22d6a63640c6
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| date | Fri, 07 Dec 2012 17:05:22 +0100 |
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1 <?xml version="1.0" encoding="utf-8"?><echo xmlns="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" xmlns:dcterms="http://purl.org/dc/terms" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:xhtml="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.0RC"> |
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2 <metadata> |
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3 <dcterms:identifier>ECHO:XT0KZ8QC.xml</dcterms:identifier> |
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4 <dcterms:creator>Harriot, Thomas</dcterms:creator> |
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5 <dcterms:title xml:lang="en">Mss. 6784</dcterms:title> |
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6 <dcterms:date xsi:type="dcterms:W3CDTF">o. J.</dcterms:date> |
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7 <dcterms:language xsi:type="dcterms:ISO639-3">eng</dcterms:language> |
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8 <dcterms:rights>CC-BY-SA</dcterms:rights> |
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9 <dcterms:license xlink:href="http://creativecommons.org/licenses/by-sa/3.0/">CC-BY-SA</dcterms:license> |
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10 <dcterms:rightsHolder xlink:href="http://www.mpiwg-berlin.mpg.de">Max Planck Institute for the History of Science, Library</dcterms:rightsHolder> |
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11 <echodir>/permanent/library/XT0KZ8QC</echodir> |
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12 <log>Automatically generated by bare_xml.py on Tue Nov 15 14:20:53 2011</log> |
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13 </metadata> |
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14 |
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15 <text xml:lang="eng" type="free"> |
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16 <div xml:id="echoid-div1" type="section" level="1" n="1"> |
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17 <pb file="add_6784_f001" o="1" n="1"/> |
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18 <head xml:id="echoid-head1" xml:space="preserve" xml:lang="lat"> |
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19 De resectione rationis |
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20 </head> |
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21 <pb file="add_6784_f001v" o="1v" n="2"/> |
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22 <pb file="add_6784_f002" o="2" n="3"/> |
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23 <head xml:id="echoid-head2" xml:space="preserve" xml:lang="lat"> |
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24 De resectione rationis |
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25 </head> |
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26 <head xml:id="echoid-head3" xml:space="preserve"> |
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27 AB) |
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28 </head> |
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29 <pb file="add_6784_f002v" o="2v" n="4"/> |
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30 <pb file="add_6784_f003" o="3" n="5"/> |
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31 <head xml:id="echoid-head4" xml:space="preserve" xml:lang="lat"> |
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32 De resectione rationis |
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33 </head> |
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34 <pb file="add_6784_f003v" o="3v" n="6"/> |
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35 <pb file="add_6784_f004" o="4" n="7"/> |
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36 <head xml:id="echoid-head5" xml:space="preserve" xml:lang="lat"> |
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37 De resectione rationis |
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38 </head> |
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39 <pb file="add_6784_f004v" o="4v" n="8"/> |
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40 <pb file="add_6784_f005" o="5" n="9"/> |
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41 <head xml:id="echoid-head6" xml:space="preserve" xml:lang="lat"> |
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42 De resectione rationis |
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43 </head> |
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44 <pb file="add_6784_f005v" o="5v" n="10"/> |
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45 <pb file="add_6784_f006" o="6" n="11"/> |
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46 <head xml:id="echoid-head7" xml:space="preserve" xml:lang="lat"> |
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47 De resectione rationis |
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48 </head> |
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49 <pb file="add_6784_f006v" o="6v" n="12"/> |
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50 <pb file="add_6784_f007" o="7" n="13"/> |
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51 <head xml:id="echoid-head8" xml:space="preserve" xml:lang="lat"> |
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52 De resectione rationis |
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53 </head> |
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54 <head xml:id="echoid-head9" xml:space="preserve"> |
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55 AB) |
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56 </head> |
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57 <pb file="add_6784_f007v" o="7v" n="14"/> |
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58 <pb file="add_6784_f008" o="8" n="15"/> |
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59 <head xml:id="echoid-head10" xml:space="preserve" xml:lang="lat"> |
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60 De resectione rationis |
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61 </head> |
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62 <pb file="add_6784_f008v" o="8v" n="16"/> |
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63 <pb file="add_6784_f009" o="9" n="17"/> |
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64 <head xml:id="echoid-head11" xml:space="preserve" xml:lang="lat"> |
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65 De resectione rationis |
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66 </head> |
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67 <head xml:id="echoid-head12" xml:space="preserve"> |
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68 2.AB) |
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69 </head> |
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70 <pb file="add_6784_f009v" o="9v" n="18"/> |
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71 <pb file="add_6784_f010" o="10" n="19"/> |
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72 <head xml:id="echoid-head13" xml:space="preserve" xml:lang="lat"> |
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73 De resectione rationis |
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74 </head> |
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75 <head xml:id="echoid-head14" xml:space="preserve"> |
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76 AC) |
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77 </head> |
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78 <pb file="add_6784_f010v" o="10v" n="20"/> |
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79 <pb file="add_6784_f011" o="11" n="21"/> |
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80 <head xml:id="echoid-head15" xml:space="preserve" xml:lang="lat"> |
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81 De resectione rationis |
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82 </head> |
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83 <head xml:id="echoid-head16" xml:space="preserve"> |
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84 AC.1) |
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85 </head> |
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86 <pb file="add_6784_f011v" o="11v" n="22"/> |
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87 <pb file="add_6784_f012" o="12" n="23"/> |
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88 <head xml:id="echoid-head17" xml:space="preserve" xml:lang="lat"> |
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89 De resectione rationis |
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90 </head> |
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91 <pb file="add_6784_f012v" o="12v" n="24"/> |
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92 <pb file="add_6784_f013" o="13" n="25"/> |
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93 <head xml:id="echoid-head18" xml:space="preserve" xml:lang="lat"> |
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94 De resectione rationis |
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95 </head> |
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96 <head xml:id="echoid-head19" xml:space="preserve"> |
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97 2.BC) |
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98 </head> |
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99 <pb file="add_6784_f013v" o="13v" n="26"/> |
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100 <pb file="add_6784_f014" o="14" n="27"/> |
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101 <head xml:id="echoid-head20" xml:space="preserve" xml:lang="lat"> |
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102 De resectione rationis |
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103 </head> |
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104 <head xml:id="echoid-head21" xml:space="preserve"> |
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105 1.BC) |
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106 </head> |
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107 <pb file="add_6784_f014v" o="14v" n="28"/> |
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108 <pb file="add_6784_f015" o="15" n="29"/> |
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109 <head xml:id="echoid-head22" xml:space="preserve" xml:lang="lat"> |
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110 De resectione rationis |
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111 </head> |
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112 <pb file="add_6784_f015v" o="15v" n="30"/> |
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113 <pb file="add_6784_f016" o="16" n="31"/> |
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114 <head xml:id="echoid-head23" xml:space="preserve" xml:lang="lat"> |
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115 De resectione rationis |
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116 </head> |
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117 <pb file="add_6784_f016v" o="16v" n="32"/> |
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118 <pb file="add_6784_f017" o="17" n="33"/> |
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119 <head xml:id="echoid-head24" xml:space="preserve" xml:lang="lat"> |
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120 Pappus 171. ad resectione rationis |
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121 </head> |
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122 <pb file="add_6784_f017v" o="17v" n="34"/> |
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123 <pb file="add_6784_f018" o="18" n="35"/> |
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124 <head xml:id="echoid-head25" xml:space="preserve" xml:lang="lat"> |
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125 De resectione rationis |
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126 </head> |
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127 <pb file="add_6784_f018v" o="18v" n="36"/> |
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128 <pb file="add_6784_f019" o="19" n="37"/> |
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129 <head xml:id="echoid-head26" xml:space="preserve" xml:lang="lat"> |
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130 De resectione spatij, problema |
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131 </head> |
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132 <head xml:id="echoid-head27" xml:space="preserve"> |
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133 a) |
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134 </head> |
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135 <pb file="add_6784_f019v" o="19v" n="38"/> |
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136 <pb file="add_6784_f020" o="20" n="39"/> |
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137 <head xml:id="echoid-head28" xml:space="preserve"> |
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138 Poristike |
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139 </head> |
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140 <pb file="add_6784_f020v" o="20v" n="40"/> |
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141 <pb file="add_6784_f021" o="21" n="41"/> |
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142 <pb file="add_6784_f021v" o="21v" n="42"/> |
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143 <pb file="add_6784_f022" o="22" n="43"/> |
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144 <pb file="add_6784_f022v" o="22v" n="44"/> |
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145 <pb file="add_6784_f023" o="23" n="45"/> |
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146 <pb file="add_6784_f023v" o="23v" n="46"/> |
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147 <pb file="add_6784_f024" o="24" n="47"/> |
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148 <head xml:id="echoid-head29" xml:space="preserve" xml:lang="lat"> |
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149 De sectione rationis |
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150 </head> |
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151 <head xml:id="echoid-head30" xml:space="preserve"> |
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152 b.1) |
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153 </head> |
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154 <pb file="add_6784_f024v" o="24v" n="48"/> |
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155 <pb file="add_6784_f025" o="25" n="49"/> |
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156 <head xml:id="echoid-head31" xml:space="preserve" xml:lang="lat"> |
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157 De sectione rationis |
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158 </head> |
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159 <head xml:id="echoid-head32" xml:space="preserve"> |
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160 b.2) |
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161 </head> |
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162 <pb file="add_6784_f025v" o="25v" n="50"/> |
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163 <pb file="add_6784_f026" o="26" n="51"/> |
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164 <head xml:id="echoid-head33" xml:space="preserve" xml:lang="lat"> |
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165 De sectione rationis |
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166 </head> |
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167 <head xml:id="echoid-head34" xml:space="preserve"> |
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168 b.3) |
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169 </head> |
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170 <pb file="add_6784_f026v" o="26v" n="52"/> |
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171 <pb file="add_6784_f027" o="27" n="53"/> |
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172 <head xml:id="echoid-head35" xml:space="preserve" xml:lang="lat"> |
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173 De sectione rationis |
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174 </head> |
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175 <head xml:id="echoid-head36" xml:space="preserve"> |
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176 b.4) |
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177 </head> |
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178 <pb file="add_6784_f027v" o="27v" n="54"/> |
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179 <pb file="add_6784_f028" o="28" n="55"/> |
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180 <head xml:id="echoid-head37" xml:space="preserve" xml:lang="lat"> |
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181 Lemma ad sectionem rationis <lb/> |
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182 et spatij |
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183 </head> |
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184 <pb file="add_6784_f028v" o="28v" n="56"/> |
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185 <pb file="add_6784_f029" o="29" n="57"/> |
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186 <pb file="add_6784_f029v" o="29v" n="58"/> |
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187 <pb file="add_6784_f030" o="30" n="59"/> |
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188 <pb file="add_6784_f030v" o="30v" n="60"/> |
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189 <pb file="add_6784_f031" o="31" n="61"/> |
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190 <pb file="add_6784_f031v" o="31v" n="62"/> |
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191 <pb file="add_6784_f032" o="32" n="63"/> |
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192 <pb file="add_6784_f032v" o="32v" n="64"/> |
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193 <pb file="add_6784_f033" o="33" n="65"/> |
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194 <pb file="add_6784_f033v" o="33v" n="66"/> |
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195 <pb file="add_6784_f034" o="34" n="67"/> |
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196 <pb file="add_6784_f034v" o="34v" n="68"/> |
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197 <pb file="add_6784_f035" o="35" n="69"/> |
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198 <pb file="add_6784_f035v" o="35v" n="70"/> |
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199 <pb file="add_6784_f036" o="36" n="71"/> |
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200 <pb file="add_6784_f036v" o="36v" n="72"/> |
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201 <pb file="add_6784_f037" o="37" n="73"/> |
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202 <pb file="add_6784_f037v" o="37v" n="74"/> |
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203 <pb file="add_6784_f038" o="38" n="75"/> |
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204 <pb file="add_6784_f038v" o="38v" n="76"/> |
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205 <pb file="add_6784_f039" o="39" n="77"/> |
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206 <pb file="add_6784_f039v" o="39v" n="78"/> |
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207 <pb file="add_6784_f040" o="40" n="79"/> |
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208 <head xml:id="echoid-head38" xml:space="preserve" xml:lang="lat"> |
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209 De resectione rationis |
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210 </head> |
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211 <pb file="add_6784_f040v" o="40v" n="80"/> |
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212 <div xml:id="echoid-div1" type="page_commentary" level="2" n="1"> |
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213 <p> |
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214 <s xml:id="echoid-s1" xml:space="preserve">[<emph style="it">Note: |
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215 <p> |
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216 <s xml:id="echoid-s1" xml:space="preserve"> |
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217 De infinitis |
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218 <lb/>[<emph style="it">tr: |
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219 On infinity |
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220 </emph>]<lb/> |
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221 </s> |
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222 </p> |
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223 </emph>] |
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224 <lb/><lb/></s></p></div> |
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225 <p xml:lang="lat"> |
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226 <s xml:id="echoid-s3" xml:space="preserve"> |
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227 Maior et Maior rationum infinitum. <lb/> |
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228 fit termini minores et minores; cum probuerit indivisibilibis <lb/> |
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229 ratio tandem infinitum. |
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230 <lb/>[<emph style="it">tr: |
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231 A greater and greater infinite ratio. |
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232 the terms are smaller and smaller; |
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233 while from indivisibles there will eventually come an infinite ratio. |
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234 </emph>]<lb/> |
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235 </s> |
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236 </p> |
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237 <p xml:lang="lat"> |
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238 <s xml:id="echoid-s4" xml:space="preserve"> |
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239 HA ad IA non potest <lb/> |
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240 esse maior BA ad BC. <lb/> |
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241 terminis scilicet decrescentibus. |
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242 <lb/>[<emph style="it">tr: |
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243 HA to IA cannot be greater than BA to BC. |
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244 the terms of course decreasing. |
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245 </emph>]<lb/> |
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246 </s> |
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247 </p> |
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248 <pb file="add_6784_f041" o="41" n="81"/> |
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249 <pb file="add_6784_f041v" o="41v" n="82"/> |
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250 <pb file="add_6784_f042" o="42" n="83"/> |
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251 <pb file="add_6784_f042v" o="42v" n="84"/> |
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252 <pb file="add_6784_f043" o="43" n="85"/> |
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253 <pb file="add_6784_f043v" o="43v" n="86"/> |
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254 <pb file="add_6784_f044" o="44" n="87"/> |
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255 <pb file="add_6784_f044v" o="44v" n="88"/> |
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256 <pb file="add_6784_f045" o="45" n="89"/> |
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257 <pb file="add_6784_f045v" o="45v" n="90"/> |
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258 <pb file="add_6784_f046" o="46" n="91"/> |
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259 <pb file="add_6784_f046v" o="46v" n="92"/> |
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260 <pb file="add_6784_f047" o="47" n="93"/> |
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261 <pb file="add_6784_f047v" o="47v" n="94"/> |
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262 <pb file="add_6784_f048" o="48" n="95"/> |
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263 <pb file="add_6784_f048v" o="48v" n="96"/> |
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264 <pb file="add_6784_f049" o="49" n="97"/> |
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265 <pb file="add_6784_f049v" o="49v" n="98"/> |
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266 <pb file="add_6784_f050" o="50" n="99"/> |
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267 <pb file="add_6784_f050v" o="50v" n="100"/> |
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268 <pb file="add_6784_f051" o="51" n="101"/> |
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269 <pb file="add_6784_f051v" o="51v" n="102"/> |
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270 <pb file="add_6784_f052" o="52" n="103"/> |
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271 <pb file="add_6784_f052v" o="52v" n="104"/> |
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272 <pb file="add_6784_f053" o="53" n="105"/> |
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273 <pb file="add_6784_f053v" o="53v" n="106"/> |
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274 <pb file="add_6784_f054" o="54" n="107"/> |
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275 <pb file="add_6784_f054v" o="54v" n="108"/> |
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276 <pb file="add_6784_f055" o="55" n="109"/> |
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277 <pb file="add_6784_f055v" o="55v" n="110"/> |
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278 <pb file="add_6784_f056" o="56" n="111"/> |
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279 <pb file="add_6784_f056v" o="56v" n="112"/> |
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280 <pb file="add_6784_f057" o="57" n="113"/> |
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281 <pb file="add_6784_f057v" o="57v" n="114"/> |
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282 <pb file="add_6784_f058" o="58" n="115"/> |
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283 <pb file="add_6784_f058v" o="58v" n="116"/> |
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284 <pb file="add_6784_f059" o="59" n="117"/> |
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285 <pb file="add_6784_f059v" o="59v" n="118"/> |
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286 <pb file="add_6784_f060" o="60" n="119"/> |
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287 <pb file="add_6784_f060v" o="60v" n="120"/> |
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288 <pb file="add_6784_f061" o="61" n="121"/> |
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289 <pb file="add_6784_f061v" o="61v" n="122"/> |
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290 <pb file="add_6784_f062" o="62" n="123"/> |
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291 <pb file="add_6784_f062v" o="62v" n="124"/> |
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292 <pb file="add_6784_f063" o="63" n="125"/> |
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293 <pb file="add_6784_f063v" o="63v" n="126"/> |
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294 <pb file="add_6784_f064" o="64" n="127"/> |
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295 <pb file="add_6784_f064v" o="64v" n="128"/> |
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296 <pb file="add_6784_f065" o="65" n="129"/> |
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297 <pb file="add_6784_f065v" o="65v" n="130"/> |
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298 <pb file="add_6784_f066" o="66" n="131"/> |
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299 <pb file="add_6784_f066v" o="66v" n="132"/> |
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300 <pb file="add_6784_f067" o="67" n="133"/> |
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301 <div xml:id="echoid-div2" type="page_commentary" level="2" n="2"> |
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302 <p> |
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303 <s xml:id="echoid-s5" xml:space="preserve">[<emph style="it">Note: |
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304 <p> |
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305 <s xml:id="echoid-s5" xml:space="preserve"> |
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306 The reference on this page is to Willebrord Snell's |
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307 <emph style="it">Apollonius Batavus</emph> (1608). |
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308 </s> |
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309 </p> |
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310 </emph>] |
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311 <lb/><lb/></s></p></div> |
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312 <head xml:id="echoid-head39" xml:space="preserve" xml:lang="lat"> |
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313 Diagrammata <lb/> |
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314 Snellij |
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315 <lb/>[<emph style="it">tr: |
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316 Snell's diagrams |
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317 </emph>]<lb/> |
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318 </head> |
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319 <pb file="add_6784_f067v" o="67v" n="134"/> |
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320 <pb file="add_6784_f068" o="68" n="135"/> |
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321 <pb file="add_6784_f068v" o="68v" n="136"/> |
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322 <pb file="add_6784_f069" o="69" n="137"/> |
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323 <pb file="add_6784_f069v" o="69v" n="138"/> |
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324 <pb file="add_6784_f070" o="70" n="139"/> |
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325 <pb file="add_6784_f070v" o="70v" n="140"/> |
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326 <pb file="add_6784_f071" o="71" n="141"/> |
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327 <pb file="add_6784_f071v" o="71v" n="142"/> |
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328 <pb file="add_6784_f072" o="72" n="143"/> |
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329 <pb file="add_6784_f072v" o="72v" n="144"/> |
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330 <pb file="add_6784_f073" o="73" n="145"/> |
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331 <pb file="add_6784_f073v" o="73v" n="146"/> |
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332 <pb file="add_6784_f074" o="74" n="147"/> |
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333 <pb file="add_6784_f074v" o="74v" n="148"/> |
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334 <pb file="add_6784_f075" o="75" n="149"/> |
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335 <pb file="add_6784_f075v" o="75v" n="150"/> |
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336 <pb file="add_6784_f076" o="76" n="151"/> |
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337 <pb file="add_6784_f076v" o="76v" n="152"/> |
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338 <pb file="add_6784_f077" o="77" n="153"/> |
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339 <pb file="add_6784_f077v" o="77v" n="154"/> |
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340 <pb file="add_6784_f078" o="78" n="155"/> |
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341 <pb file="add_6784_f078v" o="78v" n="156"/> |
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342 <pb file="add_6784_f079" o="79" n="157"/> |
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343 <pb file="add_6784_f079v" o="79v" n="158"/> |
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344 <pb file="add_6784_f080" o="80" n="159"/> |
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345 <pb file="add_6784_f080v" o="80v" n="160"/> |
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346 <pb file="add_6784_f081" o="81" n="161"/> |
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347 <pb file="add_6784_f081v" o="81v" n="162"/> |
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348 <pb file="add_6784_f082" o="82" n="163"/> |
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349 <pb file="add_6784_f082v" o="82v" n="164"/> |
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350 <pb file="add_6784_f083" o="83" n="165"/> |
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351 <pb file="add_6784_f083v" o="83v" n="166"/> |
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352 <pb file="add_6784_f084" o="84" n="167"/> |
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353 <pb file="add_6784_f084v" o="84v" n="168"/> |
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354 <pb file="add_6784_f085" o="85" n="169"/> |
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355 <pb file="add_6784_f085v" o="85v" n="170"/> |
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356 <pb file="add_6784_f086" o="86" n="171"/> |
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357 <pb file="add_6784_f086v" o="86v" n="172"/> |
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358 <pb file="add_6784_f087" o="87" n="173"/> |
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359 <pb file="add_6784_f087v" o="87v" n="174"/> |
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360 <pb file="add_6784_f088" o="88" n="175"/> |
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361 <div xml:id="echoid-div3" type="page_commentary" level="2" n="3"> |
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362 <p> |
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363 <s xml:id="echoid-s7" xml:space="preserve">[<emph style="it">Note: |
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364 <p> |
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365 <s xml:id="echoid-s7" xml:space="preserve"> |
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366 Calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>-</mo><mi>d</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mo>-</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>. |
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367 </s> |
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368 </p> |
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369 </emph>] |
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370 <lb/><lb/></s></p></div> |
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371 <pb file="add_6784_f088v" o="88v" n="176"/> |
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372 <pb file="add_6784_f089" o="89" n="177"/> |
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373 <pb file="add_6784_f089v" o="89v" n="178"/> |
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374 <div xml:id="echoid-div4" type="page_commentary" level="2" n="4"> |
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375 <p> |
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376 <s xml:id="echoid-s9" xml:space="preserve">[<emph style="it">Note: |
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377 <p> |
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378 <s xml:id="echoid-s9" xml:space="preserve"> |
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379 Calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>+</mo><mi>f</mi><mo>+</mo><mi>g</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>+</mo><mi>f</mi><mo>-</mo><mi>g</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>-</mo><mi>f</mi><mo>+</mo><mi>g</mi><mo maxsize="1">)</mo></mstyle></math>. |
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380 </s> |
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381 </p> |
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382 </emph>] |
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383 <lb/><lb/></s></p></div> |
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384 <pb file="add_6784_f090" o="90" n="179"/> |
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385 <pb file="add_6784_f090v" o="90v" n="180"/> |
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386 <pb file="add_6784_f091" o="91" n="181"/> |
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387 <pb file="add_6784_f091v" o="91v" n="182"/> |
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388 <pb file="add_6784_f092" o="92" n="183"/> |
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389 <pb file="add_6784_f092v" o="92v" n="184"/> |
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390 <pb file="add_6784_f093" o="93" n="185"/> |
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391 <pb file="add_6784_f093v" o="93v" n="186"/> |
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392 <pb file="add_6784_f094" o="94" n="187"/> |
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393 <pb file="add_6784_f094v" o="94v" n="188"/> |
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394 <pb file="add_6784_f095" o="95" n="189"/> |
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395 <pb file="add_6784_f095v" o="95v" n="190"/> |
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396 <pb file="add_6784_f096" o="96" n="191"/> |
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397 <pb file="add_6784_f096v" o="96v" n="192"/> |
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398 <pb file="add_6784_f097" o="97" n="193"/> |
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399 <pb file="add_6784_f097v" o="97v" n="194"/> |
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400 <pb file="add_6784_f098" o="98" n="195"/> |
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401 <pb file="add_6784_f098v" o="98v" n="196"/> |
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402 <pb file="add_6784_f099" o="99" n="197"/> |
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403 <pb file="add_6784_f099v" o="99v" n="198"/> |
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404 <pb file="add_6784_f100" o="100" n="199"/> |
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405 <pb file="add_6784_f100v" o="100v" n="200"/> |
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406 <pb file="add_6784_f101" o="101" n="201"/> |
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407 <pb file="add_6784_f101v" o="101v" n="202"/> |
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408 <pb file="add_6784_f102" o="102" n="203"/> |
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409 <pb file="add_6784_f102v" o="102v" n="204"/> |
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410 <pb file="add_6784_f103" o="103" n="205"/> |
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411 <pb file="add_6784_f103v" o="103v" n="206"/> |
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412 <pb file="add_6784_f104" o="104" n="207"/> |
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413 <pb file="add_6784_f104v" o="104v" n="208"/> |
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414 <pb file="add_6784_f105" o="105" n="209"/> |
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415 <p xml:lang="lat"> |
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416 <s xml:id="echoid-s11" xml:space="preserve"> |
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417 Graecia <lb/> |
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418 prævenians. <lb/> |
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419 excitans. <lb/> |
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420 vocans. <lb/> |
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421 operans. <lb/> |
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422 provens. <lb/> |
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423 comians. <lb/> |
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424 cooperans. <lb/> |
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425 adiunans. <lb/> |
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426 concomitans. <lb/> |
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427 subsequens. <lb/> |
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428 prosequens. |
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429 </s> |
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430 </p> |
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431 <pb file="add_6784_f105v" o="105v" n="210"/> |
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432 <pb file="add_6784_f106" o="106" n="211"/> |
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433 <pb file="add_6784_f106v" o="106v" n="212"/> |
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434 <pb file="add_6784_f107" o="107" n="213"/> |
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435 <pb file="add_6784_f107v" o="107v" n="214"/> |
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436 <pb file="add_6784_f108" o="108" n="215"/> |
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437 <pb file="add_6784_f108v" o="108v" n="216"/> |
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438 <pb file="add_6784_f109" o="109" n="217"/> |
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439 <pb file="add_6784_f109v" o="109v" n="218"/> |
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440 <pb file="add_6784_f110" o="110" n="219"/> |
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441 <pb file="add_6784_f110v" o="110v" n="220"/> |
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442 <pb file="add_6784_f111" o="111" n="221"/> |
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443 <pb file="add_6784_f111v" o="111v" n="222"/> |
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444 <pb file="add_6784_f112" o="112" n="223"/> |
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445 <pb file="add_6784_f112v" o="112v" n="224"/> |
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446 <pb file="add_6784_f113" o="113" n="225"/> |
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447 <pb file="add_6784_f113v" o="113v" n="226"/> |
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448 <pb file="add_6784_f114" o="114" n="227"/> |
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449 <pb file="add_6784_f114v" o="114v" n="228"/> |
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450 <pb file="add_6784_f115" o="115" n="229"/> |
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451 <pb file="add_6784_f115v" o="115v" n="230"/> |
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452 <pb file="add_6784_f116" o="116" n="231"/> |
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453 <pb file="add_6784_f116v" o="116v" n="232"/> |
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454 <pb file="add_6784_f117" o="117" n="233"/> |
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455 <pb file="add_6784_f117v" o="117v" n="234"/> |
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456 <pb file="add_6784_f118" o="118" n="235"/> |
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457 <pb file="add_6784_f118v" o="118v" n="236"/> |
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458 <pb file="add_6784_f119" o="119" n="237"/> |
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459 <pb file="add_6784_f119v" o="119v" n="238"/> |
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460 <pb file="add_6784_f120" o="120" n="239"/> |
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461 <pb file="add_6784_f120v" o="120v" n="240"/> |
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462 <pb file="add_6784_f121" o="121" n="241"/> |
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463 <pb file="add_6784_f121v" o="121v" n="242"/> |
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464 <pb file="add_6784_f122" o="122" n="243"/> |
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465 <pb file="add_6784_f122v" o="122v" n="244"/> |
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466 <pb file="add_6784_f123" o="123" n="245"/> |
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467 <div xml:id="echoid-div5" type="page_commentary" level="2" n="5"> |
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468 <p> |
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469 <s xml:id="echoid-s12" xml:space="preserve">[<emph style="it">Note: |
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470 <p> |
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471 <s xml:id="echoid-s12" xml:space="preserve"> |
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472 The references on this page are to Pappus, Book 7, |
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473 and to Giambattista Benedetti, |
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474 <emph style="it">Diversarum speculationum mathematicarum et physicarum liber</emph> (1585). |
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475 </s> |
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476 </p> |
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477 </emph>] |
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478 <lb/><lb/></s></p></div> |
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479 <p xml:lang="lat"> |
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480 <s xml:id="echoid-s14" xml:space="preserve"> |
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481 sit triangulum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mi>d</mi></mstyle></math> |
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482 <lb/>[<emph style="it">tr: |
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483 let there be a triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mi>d</mi></mstyle></math> |
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484 </emph>]<lb/> |
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485 </s> |
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486 <lb/> |
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487 <s xml:id="echoid-s15" xml:space="preserve"> |
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488 dico quod |
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489 <lb/>[<emph style="it">tr: |
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490 I say that |
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491 </emph>]<lb/> |
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492 </s> |
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493 </p> |
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494 <p xml:lang="lat"> |
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495 <s xml:id="echoid-s16" xml:space="preserve"> |
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496 sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi></mstyle></math> perpendicularis ad, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> |
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497 <lb/>[<emph style="it">tr: |
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498 let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi></mstyle></math> be perpendicular to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math> |
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499 </emph>]<lb/> |
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500 </s> |
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501 </p> |
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502 <p xml:lang="lat"> |
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503 <s xml:id="echoid-s17" xml:space="preserve"> |
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504 Unde sequitur |
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505 <lb/>[<emph style="it">tr: |
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506 whence it follows |
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507 </emph>]<lb/> |
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508 </s> |
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509 </p> |
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510 <p xml:lang="lat"> |
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511 <s xml:id="echoid-s18" xml:space="preserve"> |
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512 Vide, Pappum. lib. 7. prop: 122. pag. 235. <lb/> |
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513 et: Jo: Baptistum Benedictum pag. 362. |
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514 <lb/>[<emph style="it">tr: |
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515 See Pappus, Book 7, Proposition 122, page 235; and Johan Baptista Benedictus, page 362 |
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516 </emph>]<lb/> |
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517 </s> |
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518 </p> |
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519 <p xml:lang="lat"> |
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520 <s xml:id="echoid-s19" xml:space="preserve"> |
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521 verte |
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522 <lb/>[<emph style="it">tr: |
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523 turn over |
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524 </emph>]<lb/> |
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525 </s> |
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526 </p> |
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527 <pb file="add_6784_f123v" o="123v" n="246"/> |
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528 <pb file="add_6784_f124" o="124" n="247"/> |
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529 <pb file="add_6784_f124v" o="124v" n="248"/> |
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530 <pb file="add_6784_f125" o="125" n="249"/> |
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531 <pb file="add_6784_f125v" o="125v" n="250"/> |
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532 <pb file="add_6784_f126" o="126" n="251"/> |
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533 <pb file="add_6784_f126v" o="126v" n="252"/> |
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534 <pb file="add_6784_f127" o="127" n="253"/> |
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535 <head xml:id="echoid-head40" xml:space="preserve"> |
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536 Lemma. 1. Appol. Bat. pag. 81. |
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537 </head> |
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538 <p xml:lang="lat"> |
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539 <s xml:id="echoid-s20" xml:space="preserve"> |
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540 Sit: <lb/> |
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541 Dico quod: <lb/> |
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542 nam in utraque analogia |
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543 <lb/>[<emph style="it">tr: |
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544 Let: <lb/> |
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545 I say that: <lb/> |
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546 for in the both ratios |
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547 </emph>]<lb/> |
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548 </s> |
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549 </p> |
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550 <p xml:lang="lat"> |
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551 <s xml:id="echoid-s21" xml:space="preserve"> |
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552 Sed ita Snellius |
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553 <lb/>[<emph style="it">tr: |
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554 But it is thus in Snell. |
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555 </emph>]<lb/> |
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556 </s> |
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557 </p> |
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558 <pb file="add_6784_f127v" o="127v" n="254"/> |
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559 <pb file="add_6784_f128" o="128" n="255"/> |
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560 <pb file="add_6784_f128v" o="128v" n="256"/> |
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561 <pb file="add_6784_f129" o="129" n="257"/> |
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562 <pb file="add_6784_f129v" o="129v" n="258"/> |
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563 <pb file="add_6784_f130" o="130" n="259"/> |
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564 <pb file="add_6784_f130v" o="130v" n="260"/> |
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565 <pb file="add_6784_f131" o="131" n="261"/> |
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566 <pb file="add_6784_f131v" o="131v" n="262"/> |
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567 <pb file="add_6784_f132" o="132" n="263"/> |
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568 <pb file="add_6784_f132v" o="132v" n="264"/> |
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569 <pb file="add_6784_f133" o="133" n="265"/> |
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570 <pb file="add_6784_f133v" o="133v" n="266"/> |
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571 <pb file="add_6784_f134" o="134" n="267"/> |
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572 <pb file="add_6784_f134v" o="134v" n="268"/> |
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573 <pb file="add_6784_f135" o="135" n="269"/> |
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574 <pb file="add_6784_f135v" o="135v" n="270"/> |
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575 <pb file="add_6784_f136" o="136" n="271"/> |
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576 <pb file="add_6784_f136v" o="136v" n="272"/> |
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577 <pb file="add_6784_f137" o="137" n="273"/> |
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578 <pb file="add_6784_f137v" o="137v" n="274"/> |
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579 <pb file="add_6784_f138" o="138" n="275"/> |
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580 <pb file="add_6784_f138v" o="138v" n="276"/> |
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581 <pb file="add_6784_f139" o="139" n="277"/> |
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582 <pb file="add_6784_f139v" o="139v" n="278"/> |
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583 <pb file="add_6784_f140" o="140" n="279"/> |
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584 <pb file="add_6784_f140v" o="140v" n="280"/> |
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585 <pb file="add_6784_f141" o="141" n="281"/> |
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586 <pb file="add_6784_f141v" o="141v" n="282"/> |
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587 <pb file="add_6784_f142" o="142" n="283"/> |
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588 <pb file="add_6784_f142v" o="142v" n="284"/> |
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589 <pb file="add_6784_f143" o="143" n="285"/> |
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590 <pb file="add_6784_f143v" o="143v" n="286"/> |
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591 <pb file="add_6784_f144" o="144" n="287"/> |
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592 <pb file="add_6784_f144v" o="144v" n="288"/> |
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593 <pb file="add_6784_f145" o="145" n="289"/> |
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594 <pb file="add_6784_f145v" o="145v" n="290"/> |
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595 <pb file="add_6784_f146" o="146" n="291"/> |
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596 <pb file="add_6784_f146v" o="146v" n="292"/> |
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597 <pb file="add_6784_f147" o="147" n="293"/> |
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598 <pb file="add_6784_f147v" o="147v" n="294"/> |
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599 <pb file="add_6784_f148" o="148" n="295"/> |
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600 <pb file="add_6784_f148v" o="148v" n="296"/> |
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601 <pb file="add_6784_f149" o="149" n="297"/> |
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602 <pb file="add_6784_f149v" o="149v" n="298"/> |
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603 <div xml:id="echoid-div6" type="page_commentary" level="2" n="6"> |
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604 <p> |
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605 <s xml:id="echoid-s22" xml:space="preserve">[<emph style="it">Note: |
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606 <p> |
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607 <s xml:id="echoid-s22" xml:space="preserve"> |
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608 This page contains symbolic versions of Euclid Book II, Propositions 12 and 13: <lb/> |
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609 II.12.In obtuse-angle triangles the square on the side opposite the obtuse angle |
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610 is greater than the sum of the squares on the sides containing the obtuse angle |
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611 by twice the rectangle contained by one of the sides about the obtuse angle, |
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612 namely that on which the perpendicular falls, and the straight line cut off outside |
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613 by the perpendicular towards the obtuse angle. <lb/> |
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614 II.13. In acute-angled triangles the square on the side opposite the acute angle |
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615 is less than the sum of the squares on the sides containing the acute angle |
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616 by twice the rectangle contained by one of the sides about the acute angle, |
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617 namely that on which the perpendicular falls, and the straight line cut off within |
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618 by the perpendicular towards the acute angle. |
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619 </s> |
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620 </p> |
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621 </emph>] |
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622 <lb/><lb/></s></p></div> |
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623 <head xml:id="echoid-head41" xml:space="preserve" xml:lang="lat"> |
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624 Aliter de 12. 2<emph style="super">i</emph> Euclidis <lb/> |
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625 et 13. |
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626 <lb/>[<emph style="it">tr: |
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627 Another way for Euclid II.12 and 13. |
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628 </emph>]<lb/> |
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629 </head> |
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630 <pb file="add_6784_f150" o="150" n="299"/> |
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631 <pb file="add_6784_f150v" o="150v" n="300"/> |
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632 <pb file="add_6784_f151" o="151" n="301"/> |
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633 <pb file="add_6784_f151v" o="151v" n="302"/> |
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634 <div xml:id="echoid-div7" type="page_commentary" level="2" n="7"> |
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635 <p> |
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636 <s xml:id="echoid-s24" xml:space="preserve">[<emph style="it">Note: |
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637 <p> |
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638 <s xml:id="echoid-s24" xml:space="preserve"> |
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639 Calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>-</mo><mi>d</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mo>-</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>. |
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640 </s> |
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641 </p> |
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642 </emph>] |
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643 <lb/><lb/></s></p></div> |
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644 <pb file="add_6784_f152" o="152" n="303"/> |
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645 <pb file="add_6784_f152v" o="152v" n="304"/> |
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646 <pb file="add_6784_f153" o="153" n="305"/> |
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647 <pb file="add_6784_f153v" o="153v" n="306"/> |
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648 <pb file="add_6784_f154" o="154" n="307"/> |
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649 <pb file="add_6784_f154v" o="154v" n="308"/> |
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650 <div xml:id="echoid-div8" type="page_commentary" level="2" n="8"> |
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651 <p> |
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652 <s xml:id="echoid-s26" xml:space="preserve">[<emph style="it">Note: |
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653 <p> |
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654 <s xml:id="echoid-s26" xml:space="preserve"> |
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655 Calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>-</mo><mi>d</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>d</mi><mo>-</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>-</mo><mi>b</mi><mo maxsize="1">)</mo></mstyle></math>. |
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656 </s> |
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657 </p> |
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658 </emph>] |
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659 <lb/><lb/></s></p></div> |
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660 <pb file="add_6784_f155" o="155" n="309"/> |
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661 <pb file="add_6784_f155v" o="155v" n="310"/> |
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662 <pb file="add_6784_f156" o="156" n="311"/> |
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663 <pb file="add_6784_f156v" o="156v" n="312"/> |
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664 <pb file="add_6784_f157" o="157" n="313"/> |
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665 <pb file="add_6784_f157v" o="157v" n="314"/> |
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666 <pb file="add_6784_f158" o="158" n="315"/> |
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667 <pb file="add_6784_f158v" o="158v" n="316"/> |
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668 <pb file="add_6784_f159" o="159" n="317"/> |
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669 <pb file="add_6784_f159v" o="159v" n="318"/> |
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670 <pb file="add_6784_f160" o="160" n="319"/> |
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671 <head xml:id="echoid-head42" xml:space="preserve" xml:lang="lat"> |
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672 phys. lib.6. Cap. 1 |
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673 <lb/>[<emph style="it">tr: |
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674 Physics, Book 6, Chapter 1 |
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675 </emph>]<lb/> |
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676 </head> |
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677 <pb file="add_6784_f160v" o="160v" n="320"/> |
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678 <pb file="add_6784_f161" o="161" n="321"/> |
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679 <pb file="add_6784_f161v" o="161v" n="322"/> |
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680 <pb file="add_6784_f162" o="162" n="323"/> |
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681 <pb file="add_6784_f162v" o="162v" n="324"/> |
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682 <pb file="add_6784_f163" o="163" n="325"/> |
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683 <pb file="add_6784_f163v" o="163v" n="326"/> |
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684 <pb file="add_6784_f164" o="164" n="327"/> |
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685 <head xml:id="echoid-head43" xml:space="preserve" xml:lang="lat"> |
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686 Arist. lib. 6. Cap. 2 |
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687 <lb/>[<emph style="it">tr: |
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688 Aristotle, Book 6, Chapter 2 |
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689 </emph>]<lb/> |
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690 </head> |
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691 <pb file="add_6784_f164v" o="164v" n="328"/> |
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692 <pb file="add_6784_f165" o="165" n="329"/> |
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693 <pb file="add_6784_f165v" o="165v" n="330"/> |
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694 <pb file="add_6784_f166" o="166" n="331"/> |
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695 <div xml:id="echoid-div9" type="page_commentary" level="2" n="9"> |
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696 <p> |
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697 <s xml:id="echoid-s28" xml:space="preserve">[<emph style="it">Note: |
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698 <p> |
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699 <s xml:id="echoid-s28" xml:space="preserve"> |
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700 Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). |
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701 </s> |
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702 </p> |
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703 </emph>] |
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704 <lb/><lb/></s></p></div> |
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705 <head xml:id="echoid-head44" xml:space="preserve" xml:lang="lat"> |
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706 Residuum 5<emph style="super">a</emph> operationis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. |
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707 <lb/>[<emph style="it">tr: |
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708 The rest of the working (5) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. |
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709 </emph>]<lb/> |
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710 </head> |
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711 <pb file="add_6784_f166v" o="166v" n="332"/> |
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712 <pb file="add_6784_f167" o="167" n="333"/> |
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713 <div xml:id="echoid-div10" type="page_commentary" level="2" n="10"> |
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714 <p> |
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715 <s xml:id="echoid-s30" xml:space="preserve">[<emph style="it">Note: |
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716 <p> |
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717 <s xml:id="echoid-s30" xml:space="preserve"> |
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718 Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). |
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719 </s> |
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720 </p> |
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721 </emph>] |
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722 <lb/><lb/></s></p></div> |
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723 <head xml:id="echoid-head45" xml:space="preserve" xml:lang="lat"> |
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724 5<emph style="super">a</emph> operatio, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. |
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725 <lb/>[<emph style="it">tr: |
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726 Working (5) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> |
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727 </emph>]<lb/> |
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728 </head> |
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729 <pb file="add_6784_f167v" o="167v" n="334"/> |
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730 <pb file="add_6784_f168" o="168" n="335"/> |
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731 <div xml:id="echoid-div11" type="page_commentary" level="2" n="11"> |
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732 <p> |
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733 <s xml:id="echoid-s32" xml:space="preserve">[<emph style="it">Note: |
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734 <p> |
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735 <s xml:id="echoid-s32" xml:space="preserve"> |
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736 Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). |
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737 </s> |
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738 </p> |
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739 </emph>] |
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740 <lb/><lb/></s></p></div> |
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741 <pb file="add_6784_f168v" o="168v" n="336"/> |
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742 <pb file="add_6784_f169" o="169" n="337"/> |
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743 <div xml:id="echoid-div12" type="page_commentary" level="2" n="12"> |
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744 <p> |
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745 <s xml:id="echoid-s34" xml:space="preserve">[<emph style="it">Note: |
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746 <p> |
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747 <s xml:id="echoid-s34" xml:space="preserve"> |
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748 Calculations relating to formula (3) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). |
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749 </s> |
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750 </p> |
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751 </emph>] |
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752 <lb/><lb/></s></p></div> |
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753 <p xml:lang="lat"> |
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754 <s xml:id="echoid-s36" xml:space="preserve"> |
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755 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>d</mi></mstyle></math>. (si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi><mo>=</mo><mn>0</mn></mstyle></math>.) <lb/> |
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756 vel, cuivis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. |
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757 <lb/>[<emph style="it">tr: |
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758 or, for any <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> |
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759 </emph>]<lb/> |
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760 </s> |
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761 </p> |
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762 <p xml:lang="lat"> |
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763 <s xml:id="echoid-s37" xml:space="preserve"> |
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764 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>c</mi></mstyle></math>. cuivis. |
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765 <lb/>[<emph style="it">tr: |
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766 any |
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767 </emph>]<lb/> |
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768 </s> |
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769 </p> |
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770 <pb file="add_6784_f169v" o="169v" n="338"/> |
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771 <pb file="add_6784_f170" o="170" n="339"/> |
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772 <div xml:id="echoid-div13" type="page_commentary" level="2" n="13"> |
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773 <p> |
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774 <s xml:id="echoid-s38" xml:space="preserve">[<emph style="it">Note: |
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775 <p> |
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776 <s xml:id="echoid-s38" xml:space="preserve"> |
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777 Calculations relating to formula (3) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). |
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778 </s> |
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779 </p> |
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780 </emph>] |
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781 <lb/><lb/></s></p></div> |
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782 <pb file="add_6784_f170v" o="170v" n="340"/> |
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783 <pb file="add_6784_f171" o="171" n="341"/> |
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784 <pb file="add_6784_f171v" o="171v" n="342"/> |
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785 <div xml:id="echoid-div14" type="page_commentary" level="2" n="14"> |
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786 <p> |
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787 <s xml:id="echoid-s40" xml:space="preserve">[<emph style="it">Note: |
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788 <p> |
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789 <s xml:id="echoid-s40" xml:space="preserve"> |
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790 Calculations relating to formula (3) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). |
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791 </s> |
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792 </p> |
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793 </emph>] |
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794 <lb/><lb/></s></p></div> |
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795 <head xml:id="echoid-head46" xml:space="preserve" xml:lang="lat"> |
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796 Operatio. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>. |
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797 <lb/>[<emph style="it">tr: |
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798 Working on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> |
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799 </emph>]<lb/> |
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800 </head> |
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801 <pb file="add_6784_f172" o="172" n="343"/> |
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802 <pb file="add_6784_f172v" o="172v" n="344"/> |
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803 <div xml:id="echoid-div15" type="page_commentary" level="2" n="15"> |
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804 <p> |
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805 <s xml:id="echoid-s42" xml:space="preserve">[<emph style="it">Note: |
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806 <p> |
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807 <s xml:id="echoid-s42" xml:space="preserve"> |
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808 Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). |
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809 </s> |
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810 </p> |
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811 </emph>] |
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812 <lb/><lb/></s></p></div> |
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813 <head xml:id="echoid-head47" xml:space="preserve" xml:lang="lat"> |
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814 Residuum 3<emph style="super">a</emph> operationis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. |
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815 <lb/>[<emph style="it">tr: |
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816 The rest of the working (3) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. |
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817 </emph>]<lb/> |
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818 </head> |
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819 <p xml:lang="lat"> |
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820 <s xml:id="echoid-s44" xml:space="preserve"> |
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821 Residuum 4<emph style="super">a</emph> operationis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. |
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822 <lb/>[<emph style="it">tr: |
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823 The rest of the working (4) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. |
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824 </emph>]<lb/> |
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825 </s> |
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826 </p> |
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827 <pb file="add_6784_f173" o="173" n="345"/> |
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828 <div xml:id="echoid-div16" type="page_commentary" level="2" n="16"> |
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829 <p> |
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830 <s xml:id="echoid-s45" xml:space="preserve">[<emph style="it">Note: |
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831 <p> |
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832 <s xml:id="echoid-s45" xml:space="preserve"> |
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833 Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). |
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834 </s> |
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835 </p> |
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836 </emph>] |
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837 <lb/><lb/></s></p></div> |
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838 <head xml:id="echoid-head48" xml:space="preserve" xml:lang="lat"> |
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839 3<emph style="super">a</emph> operatio. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. |
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840 <lb/>[<emph style="it">tr: |
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841 Working (3) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> |
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842 </emph>]<lb/> |
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843 </head> |
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844 <p xml:lang="lat"> |
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845 <s xml:id="echoid-s47" xml:space="preserve"> |
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846 4<emph style="super">a</emph> operatio G. |
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847 <lb/>[<emph style="it">tr: |
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848 Working (4) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> |
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849 </emph>]<lb/> |
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850 </s> |
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851 </p> |
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852 <pb file="add_6784_f173v" o="173v" n="346"/> |
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853 <pb file="add_6784_f174" o="174" n="347"/> |
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854 <pb file="add_6784_f174v" o="174v" n="348"/> |
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855 <pb file="add_6784_f175" o="175" n="349"/> |
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856 <pb file="add_6784_f175v" o="175v" n="350"/> |
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857 <pb file="add_6784_f176" o="176" n="351"/> |
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858 <head xml:id="echoid-head49" xml:space="preserve"> |
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859 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.11 De tactibus |
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860 </head> |
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861 <p xml:lang="lat"> |
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862 <s xml:id="echoid-s48" xml:space="preserve"> |
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863 cave |
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864 <lb/>[<emph style="it">tr: |
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865 beware |
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866 </emph>]<lb/> |
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867 </s> |
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868 </p> |
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869 <p xml:lang="lat"> |
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870 <s xml:id="echoid-s49" xml:space="preserve"> |
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871 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. est centrum circuli <lb/> |
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872 circumscribentis. <lb/> |
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873 Tria traingula. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>d</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>. <lb/> |
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874 habet periferias æquales. |
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875 <lb/>[<emph style="it">tr: |
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876 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is the centre of the circumscribing circle. <lb/> |
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877 The three triangles, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math> have equal circumferences. |
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878 </emph>]<lb/> |
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879 </s> |
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880 </p> |
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881 <pb file="add_6784_f176v" o="176v" n="352"/> |
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882 <pb file="add_6784_f177" o="177" n="353"/> |
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883 <head xml:id="echoid-head50" xml:space="preserve"> |
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884 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.2 |
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885 </head> |
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886 <p xml:lang="lat"> |
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887 <s xml:id="echoid-s50" xml:space="preserve"> |
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888 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Δ</mo></mstyle></math>,<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>d</mi></mstyle></math>, latera <lb/> |
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889 <lb/>[...]<lb/> <lb/> |
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890 cuius superficies ut sequitur. |
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891 <lb/>[<emph style="it">tr: |
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892 Triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>d</mi></mstyle></math>, with sides: <lb/> |
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893 <lb/>[...]<lb/> <lb/> |
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894 whose surface is as follows. |
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895 </emph>]<lb/> |
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896 </s> |
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897 </p> |
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898 <pb file="add_6784_f177v" o="177v" n="354"/> |
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899 <pb file="add_6784_f178" o="178" n="355"/> |
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900 <head xml:id="echoid-head51" xml:space="preserve"> |
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901 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.3 |
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902 </head> |
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903 <p xml:lang="lat"> |
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904 <s xml:id="echoid-s51" xml:space="preserve"> |
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905 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Δ</mo></mstyle></math>,<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, latera <lb/> |
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906 <lb/>[...]<lb/> <lb/> |
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907 cuius superficies ut sequitur. |
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908 <lb/>[<emph style="it">tr: |
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909 Triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, with sides: <lb/> |
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910 <lb/>[...]<lb/> <lb/> |
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911 whose surface is as follows. |
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912 </emph>]<lb/> |
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913 </s> |
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914 </p> |
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915 <pb file="add_6784_f178v" o="178v" n="356"/> |
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916 <pb file="add_6784_f179" o="179" n="357"/> |
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917 <head xml:id="echoid-head52" xml:space="preserve"> |
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918 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.4 |
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919 </head> |
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920 <pb file="add_6784_f179v" o="179v" n="358"/> |
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921 <pb file="add_6784_f180" o="180" n="359"/> |
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922 <div xml:id="echoid-div17" type="page_commentary" level="2" n="17"> |
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923 <p> |
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924 <s xml:id="echoid-s52" xml:space="preserve">[<emph style="it">Note: |
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925 <p> |
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926 <s xml:id="echoid-s52" xml:space="preserve"> |
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927 The reference in the top right hand corner is to Viète, |
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928 <emph style="it">Apollonius Gallus</emph> (1600), Problem IX. |
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929 </s> |
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930 <lb/> |
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931 <quote xml:lang="lat"> |
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932 Problema IX. <lb/> |
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933 Datis duobus circulis, & puncto, per datum punctum circulum describere |
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934 quem duo dati circuli contingat. |
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935 </quote> |
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936 <lb/> |
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937 <quote> |
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938 IX. Given two circles and a point, through the given point describe a circle that touches the two given circles. |
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939 </quote> |
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940 </p> |
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941 </emph>] |
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942 <lb/><lb/></s></p></div> |
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943 <head xml:id="echoid-head53" xml:space="preserve"> |
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944 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.5) |
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945 </head> |
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946 <p xml:lang="lat"> |
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947 <s xml:id="echoid-s54" xml:space="preserve"> |
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948 Vide: Appol. Gall. prob. 9. |
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949 <lb/>[<emph style="it">tr: |
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950 See Apollonius Gallus, Problem IX. |
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951 </emph>]<lb/> |
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952 </s> |
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953 </p> |
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954 <p xml:lang="lat"> |
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955 <s xml:id="echoid-s55" xml:space="preserve"> |
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956 Aberratur de modo contingendi <lb/> |
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957 circulos posititios alias operatio bona <lb/> |
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958 vide igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.5.2<emph style="super">o</emph>. |
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959 <lb/>[<emph style="it">tr: |
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960 There is an error in the method of touching the supposed circles, othersie the working is good; |
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961 therefore see shee <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>..5.2. |
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962 </emph>]<lb/> |
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963 [<emph style="it">Note: |
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964 Sheet <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.5.2 is Add MS 6784, f. 181. |
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965 </emph>]<lb/> |
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966 </s> |
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967 </p> |
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968 <p xml:lang="lat"> |
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969 <s xml:id="echoid-s56" xml:space="preserve"> |
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970 radius circuli posititij (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>) minoris <lb/> |
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971 posititij (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>) maioris <lb/> |
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972 distantia centrorum |
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973 <lb/>[<emph style="it">tr: |
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974 radius of the smaller supposed circle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <lb/> |
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975 of the greater supposed circle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <lb/> |
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976 distance of the centres. |
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977 </emph>]<lb/> |
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978 </s> |
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979 </p> |
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980 <pb file="add_6784_f180v" o="180v" n="360"/> |
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981 <pb file="add_6784_f181" o="181" n="361"/> |
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982 <div xml:id="echoid-div18" type="page_commentary" level="2" n="18"> |
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983 <p> |
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984 <s xml:id="echoid-s57" xml:space="preserve">[<emph style="it">Note: |
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985 <p> |
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986 <s xml:id="echoid-s57" xml:space="preserve"> |
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987 A continuation of the work on Add MS 6784, f. 180. |
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988 </s> |
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989 </p> |
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990 </emph>] |
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991 <lb/><lb/></s></p></div> |
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992 <head xml:id="echoid-head54" xml:space="preserve"> |
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993 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.5.2<emph style="super">o</emph>) |
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994 </head> |
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995 <p xml:lang="lat"> |
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996 <s xml:id="echoid-s59" xml:space="preserve"> |
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997 Vide: Appol: Gall. prob. 9. <lb/> |
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998 fig: 2. |
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999 <lb/>[<emph style="it">tr: |
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1000 See Apollonius Gallus, Problem IX, figure 2. |
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1001 </emph>]<lb/> |
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1002 </s> |
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1003 </p> |
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1004 <p xml:lang="lat"> |
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1005 <s xml:id="echoid-s60" xml:space="preserve"> |
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1006 radius circuli posititij (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>) minoris <lb/> |
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1007 posititij (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>) maioris <lb/> |
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1008 distantia centrorum |
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1009 <lb/>[<emph style="it">tr: |
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1010 radius of the smaller supposed circle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <lb/> |
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1011 of the greater supposed circle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <lb/> |
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1012 distance of the centres. |
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1013 </emph>]<lb/> |
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1014 </s> |
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1015 </p> |
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1016 <pb file="add_6784_f181v" o="181v" n="362"/> |
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1017 <pb file="add_6784_f182" o="182" n="363"/> |
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1018 <head xml:id="echoid-head55" xml:space="preserve"> |
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1019 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.6.) |
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1020 </head> |
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1021 <p xml:lang="lat"> |
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1022 <s xml:id="echoid-s61" xml:space="preserve"> |
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1023 radius circuli posititij (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>) <lb/> |
|
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1024 posititij (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>) <lb/> |
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1025 distantia centrorum |
|
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|
1026 <lb/>[<emph style="it">tr: |
|
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|
1027 radius of the supposed circle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <lb/> |
|
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1028 of the supposed circle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <lb/> |
|
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|
1029 distance of the centres. |
|
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|
1030 </emph>]<lb/> |
|
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|
1031 </s> |
|
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|
1032 </p> |
|
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|
1033 <pb file="add_6784_f182v" o="182v" n="364"/> |
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|
1034 <pb file="add_6784_f183" o="183" n="365"/> |
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1035 <head xml:id="echoid-head56" xml:space="preserve"> |
|
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|
1036 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi></mstyle></math>.1 |
|
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|
1037 </head> |
|
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|
1038 <p xml:lang="lat"> |
|
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|
1039 <s xml:id="echoid-s62" xml:space="preserve"> |
|
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|
1040 data <lb/> |
|
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|
1041 <lb/>[...]<lb/> <lb/> |
|
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|
1042 Quæritur: vel. |
|
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|
1043 <lb/>[<emph style="it">tr: |
|
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|
1044 given <lb/> |
|
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|
1045 <lb/>[...]<lb/> <lb/> |
|
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|
1046 Sought, either: |
|
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|
1047 </emph>]<lb/> |
|
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|
1048 </s> |
|
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|
1049 </p> |
|
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|
1050 <p xml:lang="lat"> |
|
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|
1051 <s xml:id="echoid-s63" xml:space="preserve"> |
|
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|
1052 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Δ</mo></mstyle></math>,<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>d</mi></mstyle></math>, latera <lb/> |
|
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|
1053 <lb/>[...]<lb/> <lb/> |
|
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|
1054 cuius superficies ut sequitur. |
|
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|
1055 <lb/>[<emph style="it">tr: |
|
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|
1056 Triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>d</mi></mstyle></math>, with sides: <lb/> |
|
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|
1057 <lb/>[...]<lb/> <lb/> |
|
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|
1058 whose surface is as follows. |
|
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|
1059 </emph>]<lb/> |
|
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|
1060 </s> |
|
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|
1061 </p> |
|
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|
1062 <pb file="add_6784_f183v" o="183v" n="366"/> |
|
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|
1063 <pb file="add_6784_f184" o="184" n="367"/> |
|
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|
1064 <head xml:id="echoid-head57" xml:space="preserve"> |
|
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|
1065 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi></mstyle></math>.3 |
|
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|
1066 </head> |
|
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|
1067 <p xml:lang="lat"> |
|
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|
1068 <s xml:id="echoid-s64" xml:space="preserve"> |
|
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|
1069 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Δ</mo></mstyle></math>,<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mi>d</mi></mstyle></math>, latera <lb/> |
|
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|
1070 <lb/>[...]<lb/> <lb/> |
|
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|
1071 cuius superficies ut sequitur. |
|
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|
1072 <lb/>[<emph style="it">tr: |
|
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|
1073 Triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mi>d</mi></mstyle></math>, with sides: <lb/> |
|
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|
1074 <lb/>[...]<lb/> <lb/> |
|
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|
1075 whose surface is as follows. |
|
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|
1076 </emph>]<lb/> |
|
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|
1077 </s> |
|
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|
1078 </p> |
|
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|
1079 <pb file="add_6784_f184v" o="184v" n="368"/> |
|
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|
1080 <pb file="add_6784_f185" o="185" n="369"/> |
|
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|
1081 <head xml:id="echoid-head58" xml:space="preserve"> |
|
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|
1082 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi></mstyle></math>.2 |
|
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|
1083 </head> |
|
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|
1084 <p xml:lang="lat"> |
|
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|
1085 <s xml:id="echoid-s65" xml:space="preserve"> |
|
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|
1086 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Δ</mo></mstyle></math>,<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>, latera <lb/> |
|
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|
1087 <lb/>[...]<lb/> <lb/> |
|
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|
1088 cuius superficies ut sequitur. |
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1089 <lb/>[<emph style="it">tr: |
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1090 Triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>, with sides: <lb/> |
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1091 <lb/>[...]<lb/> <lb/> |
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1092 whose surface is as follows. |
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1093 </emph>]<lb/> |
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1094 </s> |
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1095 </p> |
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1096 <pb file="add_6784_f185v" o="185v" n="370"/> |
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1097 <pb file="add_6784_f186" o="186" n="371"/> |
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1098 <head xml:id="echoid-head59" xml:space="preserve"> |
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1099 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.1 De tactibus |
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1100 </head> |
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1101 <pb file="add_6784_f186v" o="186v" n="372"/> |
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1102 <pb file="add_6784_f187" o="187" n="373"/> |
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1103 <pb file="add_6784_f187v" o="187v" n="374"/> |
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1104 <pb file="add_6784_f188" o="188" n="375"/> |
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1105 <pb file="add_6784_f188v" o="188v" n="376"/> |
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1106 <pb file="add_6784_f189" o="189" n="377"/> |
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1107 <pb file="add_6784_f189v" o="189v" n="378"/> |
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1108 <pb file="add_6784_f190" o="190" n="379"/> |
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1109 <pb file="add_6784_f190v" o="190v" n="380"/> |
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1110 <pb file="add_6784_f191" o="191" n="381"/> |
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1111 <pb file="add_6784_f191v" o="191v" n="382"/> |
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1112 <pb file="add_6784_f192" o="192" n="383"/> |
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1113 <pb file="add_6784_f192v" o="192v" n="384"/> |
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1114 <pb file="add_6784_f193" o="193" n="385"/> |
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1115 <pb file="add_6784_f193v" o="193v" n="386"/> |
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1116 <pb file="add_6784_f194" o="194" n="387"/> |
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1117 <head xml:id="echoid-head60" xml:space="preserve"> |
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1118 7. (o o) |
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1119 </head> |
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1120 <pb file="add_6784_f194v" o="194v" n="388"/> |
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1121 <pb file="add_6784_f195" o="195" n="389"/> |
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1122 <head xml:id="echoid-head61" xml:space="preserve"> |
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1123 De tactibus <lb/> |
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1124 Probl. 6 (. o -) |
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1125 </head> |
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1126 <pb file="add_6784_f195v" o="195v" n="390"/> |
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1127 <pb file="add_6784_f196" o="196" n="391"/> |
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1128 <pb file="add_6784_f196v" o="196v" n="392"/> |
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1129 <pb file="add_6784_f197" o="197" n="393"/> |
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1130 <head xml:id="echoid-head62" xml:space="preserve"> |
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1131 6) De tactibus |
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1132 <lb/>[<emph style="it">tr: |
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1133 On touching |
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1134 </emph>]<lb/> |
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1135 </head> |
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1136 <p> |
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1137 <s xml:id="echoid-s66" xml:space="preserve"> |
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1138 problema. <lb/> |
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1139 Datis tribus circulis <lb/> |
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1140 sese mutuo contingentibus: <lb/> |
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1141 invenire quartum circulum <lb/> |
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1142 qui mutus tangetur in datis. |
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1143 <lb/>[<emph style="it">tr: |
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1144 Problem. <lb/> |
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1145 Given three circles, mutually touching, to find a fourth circle that is mutually touched by those given. |
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1146 </emph>]<lb/> |
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1147 </s> |
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1148 </p> |
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1149 <p> |
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1150 <s xml:id="echoid-s67" xml:space="preserve"> |
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1151 Sint tres dati circuli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi><mi>d</mi></mstyle></math>, <lb/> |
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1152 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>t</mi><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>u</mi><mi>e</mi></mstyle></math>, sese mutuo contingentes <lb/> |
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1153 in punctis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. cuius centra <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/> |
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1154 Agatur recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi></mstyle></math> in continuum <lb/> |
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1155 <lb/>[...]<lb/> <lb/> |
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1156 Agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> recta contingens <lb/> |
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1157 circulum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi><mi>d</mi></mstyle></math> in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <lb/> |
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1158 Agatur recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>z</mi></mstyle></math> in continuum quæ secabit <lb/> |
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1159 circulum cuius centrum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> puncto. <lb/> |
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1160 fiat, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>a</mi><mi>i</mi></mstyle></math> recta, parallela <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>z</mi></mstyle></math>. <lb/> |
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1161 Et ad lineam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>z</mi></mstyle></math> productam sint per-<lb/> |
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1162 pendicularis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>q</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>l</mi></mstyle></math>. |
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1163 <lb/>[<emph style="it">tr: |
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1164 Let the three given circles be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>t</mi><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>u</mi><mi>e</mi></mstyle></math>, mutually touching at the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, |
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1165 whose centres are <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/> |
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1166 There is constructed the extended line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi></mstyle></math>. <lb/> |
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1167 <lb/>[...]<lb/> <lb/> |
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1168 There is constructed the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> touching the circle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi><mi>d</mi></mstyle></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <lb/> |
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1169 There is constructed the extended line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>z</mi></mstyle></math> which will cut the circule whose centre is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> in the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. <lb/> |
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1170 Let the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>a</mi><mi>i</mi></mstyle></math>be parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>z</mi></mstyle></math>. <lb/> |
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1171 And to the extended line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>z</mi></mstyle></math> let there be perpendiculars <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>q</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>l</mi></mstyle></math>. |
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1172 </emph>]<lb/> |
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1173 </s> |
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1174 </p> |
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1175 <p> |
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1176 <s xml:id="echoid-s68" xml:space="preserve"> |
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1177 Bissecetur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. <lb/> |
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1178 Centro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, intervallo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi><mi>b</mi></mstyle></math>, <lb/> |
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1179 describatur circulus. <lb/> |
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1180 Dico quod: ille est circulus quæsitus <lb/> |
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1181 et contingit tres datos <lb/> |
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1182 in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. |
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1183 <lb/>[<emph style="it">tr: |
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1184 Let the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> be bisected at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. <lb/> |
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1185 With centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi><mi>b</mi></mstyle></math>, there is drawn a circle. <lb/> |
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1186 I say that this is the circle sought, and that it touches the tree given circles at the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. |
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1187 </emph>]<lb/> |
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1188 </s> |
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1189 </p> |
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1190 <pb file="add_6784_f197v" o="197v" n="394"/> |
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1191 <pb file="add_6784_f198" o="198" n="395"/> |
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1192 <head xml:id="echoid-head63" xml:space="preserve"> |
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1193 <emph style="st">6.)</emph> 7.) |
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1194 </head> |
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1195 <p xml:lang="lat"> |
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1196 <s xml:id="echoid-s69" xml:space="preserve"> |
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1197 Sint tres dati circuli, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi><mi>d</mi></mstyle></math>, <lb/> |
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1198 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>u</mi><mi>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi><mi>c</mi></mstyle></math>, sese mutuo <lb/> |
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1199 contingentes in punctis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, <lb/> |
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1200 cuius centra, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. |
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1201 <lb/>[<emph style="it">tr: |
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1202 Let there be three given circles, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>u</mi><mi>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi><mi>c</mi></mstyle></math>, mutually touching in the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, |
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1203 whose centres are at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. |
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1204 </emph>]<lb/> |
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1205 </s> |
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1206 </p> |
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1207 <p xml:lang="lat"> |
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1208 <s xml:id="echoid-s70" xml:space="preserve"> |
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1209 Oportet invenire circulum <lb/> |
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1210 contingentem tres datos: <lb/> |
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1211 (nempe, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>h</mi><mi>t</mi></mstyle></math>, cius centrum, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>) |
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1212 <lb/>[<emph style="it">tr: |
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1213 One must find the circle touching the three given ones (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>h</mi><mi>t</mi></mstyle></math>, with centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>). |
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1214 </emph>]<lb/> |
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1215 </s> |
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1216 </p> |
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1217 <p xml:lang="lat"> |
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1218 <s xml:id="echoid-s71" xml:space="preserve"> |
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1219 Per centra <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, agatur recta <lb/> |
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1220 et continuetur ad utraque partes <lb/> |
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1221 et fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. <lb/> |
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1222 Et ad illam lineam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>m</mi></mstyle></math> <lb/> |
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1223 perpendicularis. <lb/> |
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1224 Continuetur ad partes contrarias <lb/> |
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1225 usque ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math>, et fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mi>k</mi><mo>=</mo><mi>s</mi><mi>a</mi></mstyle></math>. <lb/> |
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1226 Tum primo, agatur recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>s</mi></mstyle></math> <lb/> |
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1227 quæ secabit periferiam circuli <lb/> |
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1228 cuius centrum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. <lb/> |
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1229 Secundo, agatur recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>k</mi></mstyle></math> <lb/> |
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1230 quæ secabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math> productam in <lb/> |
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1231 puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. <lb/> |
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1232 Ultimo, centro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, intervallo <lb/> |
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1233 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>h</mi></mstyle></math> describatur circulus. <lb/> |
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1234 Dico quod: ille est circulus quæsitus <lb/> |
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1235 et contingit tres datos in <lb/> |
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1236 punctis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>. |
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1237 <lb/>[<emph style="it">tr: |
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1238 Through the centres <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, a line is drawn and continued on both sides, and so there are |
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1239 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. <lb/> |
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1240 And to that line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>m</mi></mstyle></math> be perpendicular. <lb/> |
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1241 It is continued to both sides as far as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math>, and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mi>k</mi><mo>=</mo><mi>s</mi><mi>a</mi></mstyle></math>. <lb/> |
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1242 Then, first, there is drawn the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>s</mi></mstyle></math>, |
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1243 which will cut the circumference of the circle with centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> in the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. <lb/> |
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1244 Second, there is drawn the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>l</mi></mstyle></math>, |
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1245 which will cut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math> extended, in the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. <lb/> |
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1246 Finally, with centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> and radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>h</mi></mstyle></math>, there is drawn the required circle. <lb/> |
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1247 I say that this is the circle sought, and it touches the three given at the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>. |
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1248 </emph>]<lb/> |
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1249 </s> |
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1250 </p> |
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1251 <head xml:id="echoid-head64" xml:space="preserve" xml:lang="lat"> |
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1252 Exegesis arithmetica <lb/> |
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1253 pro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>h</mi></mstyle></math> radio. |
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1254 <lb/>[<emph style="it">tr: |
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1255 Arithmetical exegesis, for radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>h</mi></mstyle></math>. |
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1256 </emph>]<lb/> |
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1257 </head> |
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1258 <p xml:lang="lat"> |
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1259 <s xml:id="echoid-s72" xml:space="preserve"> |
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1260 Datorum circulorum radii <lb/> |
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1261 dati sunt, et centrorum <lb/> |
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1262 distantiæ. <lb/> |
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1263 Ergo lateri trianguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>a</mi><mi>y</mi></mstyle></math> <lb/> |
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1264 data sunt. Inde perpendicularis <lb/> |
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1265 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>m</mi></mstyle></math>, et recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>m</mi></mstyle></math>. Inde tota <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>m</mi></mstyle></math>. <lb/> |
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1266 Inde datur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math>. Inde <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>θ</mi></mstyle></math>. <lb/> |
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1267 Tum cum datur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mi>a</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>m</mi></mstyle></math>, datur <lb/> |
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1268 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mi>m</mi></mstyle></math> et inde <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>s</mi></mstyle></math>. Et cum datur <lb/> |
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1269 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>θ</mi></mstyle></math>, datur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>s</mi></mstyle></math>. <lb/> |
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1270 Tum lineæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> ad angulos <lb/> |
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1271 rectos et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi></mstyle></math> pro-<lb/> |
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1272 ducta concurret cum illa <lb/> |
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1273 in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>h</mi></mstyle></math> sunt <lb/> |
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1274 æquales. et triangulum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi><mi>h</mi></mstyle></math> <lb/> |
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1275 simile est triangulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>s</mi><mi>h</mi></mstyle></math>, <lb/> |
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1276 cuius latera data sunt. et <lb/> |
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1277 antea datum fuit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math>. ergo dantur <lb/> |
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1278 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>h</mi></mstyle></math>. <lb/> |
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1279 <lb/>[...]<lb/> <lb/> |
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1280 Ergo tota <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>a</mi></mstyle></math> datur <lb/> |
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1281 <lb/>[...]<lb/> <lb/> |
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1282 Ergo datur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi></mstyle></math> <lb/> |
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1283 sed antea nota fuit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>, <lb/> |
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1284 ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>p</mi></mstyle></math> datur <lb/> |
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1285 Quod quærebatur. |
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1286 <lb/>[<emph style="it">tr: |
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1287 The radii of the fiven circles are given, and the distances of their centres. <lb/> |
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1288 Therefore the sides of the triangles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>a</mi><mi>y</mi></mstyle></math> are given. |
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1289 Hence the perpendicular <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>m</mi></mstyle></math>, and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>m</mi></mstyle></math>. Hence the total, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>m</mi></mstyle></math>. |
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1290 Hence there is given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math>. Hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>θ</mi></mstyle></math>. |
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1291 Then since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>m</mi></mstyle></math> are given, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mi>m</mi></mstyle></math> is given and thence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>s</mi></mstyle></math>. |
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1292 And since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>θ</mi></mstyle></math> are given, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>s</mi></mstyle></math> are given. <lb/> |
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1293 Then the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> is at right angles to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi></mstyle></math> extended meets with it at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. |
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1294 The lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>h</mi></mstyle></math> are equal. And the triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi><mi>h</mi></mstyle></math> is similar to triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>s</mi><mi>h</mi></mstyle></math>, |
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1295 whose sides are given. And earlier <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math> was given. Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>h</mi></mstyle></math> are given. <lb/> |
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1296 <lb/>[...]<lb/> <lb/> |
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1297 Therefore the total <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>a</mi></mstyle></math> is given. <lb/> |
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1298 <lb/>[...]<lb/> <lb/> |
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1299 Therfore there is given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi></mstyle></math>, but earlier <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math> became known, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>p</mi></mstyle></math> is given. <lb/> |
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1300 Which was sought. |
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1301 </emph>]<lb/> |
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1302 </s> |
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1303 </p> |
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1304 <p xml:lang="lat"> |
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1305 <s xml:id="echoid-s73" xml:space="preserve"> |
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1306 Per doctrinam sinuum <lb/> |
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1307 opus abbreviatur, sed <lb/> |
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1308 alia method ut convenit. |
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1309 <lb/>[<emph style="it">tr: |
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1310 By the doctrine of sines, the work is shorter, but another method, as convenient. |
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1311 </emph>]<lb/> |
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1312 </s> |
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1313 </p> |
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1314 <pb file="add_6784_f198v" o="198v" n="396"/> |
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1315 <pb file="add_6784_f199" o="199" n="397"/> |
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1316 <pb file="add_6784_f199v" o="199v" n="398"/> |
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1317 <pb file="add_6784_f200" o="200" n="399"/> |
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1318 <head xml:id="echoid-head65" xml:space="preserve"> |
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1319 6.) |
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1320 </head> |
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1321 <head xml:id="echoid-head66" xml:space="preserve" xml:lang="lat"> |
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1322 Arithmetica Exegesis <lb/> |
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1323 radij <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>y</mi></mstyle></math> |
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1324 <lb/>[<emph style="it">tr: |
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1325 Arithmetical exegesis, for radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>y</mi></mstyle></math>. |
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1326 </emph>]<lb/> |
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1327 </head> |
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1328 <p xml:lang="lat"> |
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1329 <s xml:id="echoid-s74" xml:space="preserve"> |
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1330 Datorum circulorum radij dati <lb/> |
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1331 sunt, et centrorum distantiæ <lb/> |
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1332 Ergo lateri trianguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <lb/> |
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1333 cum sit, ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi><mo>,</mo><mi>h</mi><mi>p</mi><mo>:</mo><mi>a</mi><mi>f</mi><mo>,</mo><mi>f</mi><mi>p</mi></mstyle></math>. <lb/> |
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1334 datur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>p</mi></mstyle></math>. et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>h</mi></mstyle></math> cui æqualis <lb/> |
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1335 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> contingens. |
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1336 <lb/>[<emph style="it">tr: |
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1337 The radii of given circles are given, and the distances of their centres. <lb/> |
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1338 Therefore the sides of the triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>p</mi><mi>a</mi></mstyle></math>, and since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi><mo>:</mo><mi>h</mi><mi>p</mi><mo>=</mo><mi>a</mi><mi>f</mi><mo>:</mo><mi>f</mi><mi>p</mi></mstyle></math>, there is given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>p</mi></mstyle></math>, |
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1339 and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>h</mi></mstyle></math>, which is equal to the angent <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math>. |
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1340 </emph>]<lb/> |
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1341 </s> |
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1342 </p> |
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1343 <p xml:lang="lat"> |
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1344 <s xml:id="echoid-s75" xml:space="preserve"> |
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1345 Ex <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>z</mi></mstyle></math> datis, datur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>z</mi></mstyle></math>. <lb/> |
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1346 Sunt igitur duo triangula <lb/> |
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1347 datorum laterum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>p</mi><mi>z</mi></mstyle></math>. <lb/> |
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1348 constituuntur super eandem <lb/> |
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1349 basim <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>z</mi></mstyle></math>. datur igitur verti-<lb/> |
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1350 cum distantia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>b</mi></mstyle></math>. |
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1351 <lb/>[<emph style="it">tr: |
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1352 From <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>z</mi></mstyle></math>, given, there is given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>z</mi></mstyle></math>. <lb/> |
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1353 Therefore there are two triangles with given sides <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>p</mi><mi>z</mi></mstyle></math>, constructed on the same base <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>z</mi></mstyle></math>. <lb/> |
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1354 Therefore the vertical distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>b</mi></mstyle></math> is given. |
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1355 </emph>]<lb/> |
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1356 </s> |
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1357 </p> |
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1358 <p xml:lang="lat"> |
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1359 <s xml:id="echoid-s76" xml:space="preserve"> |
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1360 Ex triangulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>p</mi><mi>z</mi></mstyle></math> datorum laterum <lb/> |
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1361 datur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>n</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi></mstyle></math> perpendicularis <lb/> |
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1362 nota igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>n</mi></mstyle></math>. <lb/> |
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1363 fiunt <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>η</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>λ</mi></mstyle></math>, æquales radio <lb/> |
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1364 circuli circa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. <lb/> |
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1365 Dantur, igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>η</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>λ</mi></mstyle></math>. <lb/> |
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1366 Tum: <lb/> |
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1367 Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, cuius dimidium <lb/> |
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1368 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>y</mi></mstyle></math>, radius quæsitus. |
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1369 <lb/>[<emph style="it">tr: |
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1370 From the triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>p</mi><mi>z</mi></mstyle></math> with given sides there is given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>n</mi></mstyle></math>, |
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1371 and the perpendicular <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi></mstyle></math> is known, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>n</mi></mstyle></math>. <lb/> |
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1372 There are constructed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>η</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>λ</mi></mstyle></math>, equal to the radius of the circle about <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. <lb/> |
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1373 Therefore there are given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>η</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>λ</mi></mstyle></math>. <lb/> |
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1374 Then: <lb/> |
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1375 Therefore there is given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, whose half, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>y</mi></mstyle></math>, is the sought radius. <lb/> |
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1376 </emph>]<lb/> |
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1377 </s> |
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1378 </p> |
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1379 <p xml:lang="lat"> |
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1380 <s xml:id="echoid-s77" xml:space="preserve"> |
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1381 Per Canonem triangulorum <lb/> |
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1382 alia methodo <emph style="super">ut covenit</emph>, operatio fit <lb/> |
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1383 brevior. |
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1384 <lb/>[<emph style="it">tr: |
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1385 By the Canons for triangles, there is another method, as convenient, which may be carried ore briefly. |
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1386 </emph>]<lb/> |
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1387 </s> |
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1388 </p> |
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1389 <p xml:lang="lat"> |
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1390 <s xml:id="echoid-s78" xml:space="preserve"> |
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1391 Nota. <lb/> |
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1392 per puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>η</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>λ</mi></mstyle></math> <lb/> |
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1393 fit etiam geometrica <lb/> |
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1394 constructio, loco <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi></mstyle></math>. |
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1395 <lb/>[<emph style="it">tr: |
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1396 Note. <lb/> |
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1397 Through the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>η</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>λ</mi></mstyle></math> there may also be carried out a geometric construction, instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi></mstyle></math>. |
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1398 </emph>]<lb/> |
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1399 </s> |
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1400 </p> |
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1401 <head xml:id="echoid-head67" xml:space="preserve" xml:lang="lat"> |
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1402 Arithmetica exegesis <lb/> |
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1403 radij <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math> <lb/> |
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1404 cæteris datis. |
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1405 <lb/>[<emph style="it">tr: |
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1406 Arithmetical exegesis, for radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>, given the rest. |
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1407 </emph>]<lb/> |
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1408 </head> |
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1409 <p xml:lang="lat"> |
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1410 <s xml:id="echoid-s79" xml:space="preserve"> |
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1411 Datorum circulorum radij dati <lb/> |
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1412 sunt, et centrorum distantiæ <lb/> |
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1413 Ergo lateri trianguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, <lb/> |
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1414 Datur igitur perpendicularis <lb/> |
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1415 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi></mstyle></math>, et linea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>n</mi></mstyle></math>. Unde nota <lb/> |
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1416 fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>p</mi></mstyle></math>. <lb/> |
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1417 Cum data <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>o</mi></mstyle></math> <lb/> |
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1418 unde data <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math>. <lb/> |
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1419 Tum, trianguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>p</mi><mi>o</mi></mstyle></math> latera sunt <lb/> |
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1420 nota; unde nota perpendicularis <lb/> |
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1421 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>u</mi></mstyle></math>. Et linea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>u</mi></mstyle></math>, cui æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>h</mi></mstyle></math>. <lb/> |
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1422 Dantur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>h</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math>. <lb/> |
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1423 Dantur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>f</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>f</mi></mstyle></math>. <lb/> |
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1424 Denique fiat: <lb/> |
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1425 Datur igiture <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>, quod <lb/> |
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1426 quærebatur. |
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1427 <lb/>[<emph style="it">tr: |
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1428 The radii of given circles are given, and the distances of their centres. <lb/> |
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1429 Therefore the sides of the triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>p</mi><mi>y</mi></mstyle></math>. <lb/> |
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1430 Therefore there is given the perpendicular <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi></mstyle></math>, and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>n</mi></mstyle></math>. Whence there is known <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>p</mi></mstyle></math>. <lb/> |
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1431 Since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>o</mi></mstyle></math> are given, there is given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math>. <lb/> |
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1432 Then the sides of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>p</mi><mi>o</mi></mstyle></math> are known, whence the perpendicular <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>u</mi></mstyle></math> is known. |
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1433 And the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>u</mi></mstyle></math>, which is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>h</mi></mstyle></math>. <lb/> |
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1434 Therefore there are given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>h</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math>. <lb/> |
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1435 Thereofre there are given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>f</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>f</mi></mstyle></math>. <lb/> |
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1436 Then let there be constructed: <lb/> |
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1437 Therefore there is given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>, which was sought. |
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1438 </emph>]<lb/> |
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1439 </s> |
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1440 </p> |
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1441 <head xml:id="echoid-head68" xml:space="preserve" xml:lang="lat"> |
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1442 Geometria exegesis <lb/> |
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1443 ipsius radii <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>. |
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1444 <lb/>[<emph style="it">tr: |
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1445 Geometric exegesis, for the same radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>. |
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1446 </emph>]<lb/> |
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1447 </head> |
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1448 <p xml:lang="lat"> |
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1449 <s xml:id="echoid-s80" xml:space="preserve"> |
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1450 Trium datorum circulorum <lb/> |
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1451 centra <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, connectantur. <lb/> |
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1452 per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math> fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> acta <lb/> |
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1453 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> faciat angulos rectos cum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. <lb/> |
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1454 Ita <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi></mstyle></math>; quæ secabit circulum <lb/> |
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1455 circa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. <lb/> |
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1456 Agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math>, quæ producta secabit <lb/> |
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1457 eandem circulum circa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. <lb/> |
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1458 Agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>p</mi></mstyle></math> et producatur ad <lb/> |
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1459 utraque partes quæ secabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> <lb/> |
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1460 in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <lb/> |
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1461 Tum fiat: <lb/> |
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1462 Datur igiture <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>a</mi></mstyle></math>, et centrum circuli <lb/> |
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1463 quæsiti. |
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1464 <lb/>[<emph style="it">tr: |
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1465 Let the centres of the given circles, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, be connected. <lb/> |
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1466 Through <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math> let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> be constructed; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> makes a right angle with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. <lb/> |
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1467 Thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi></mstyle></math>, which cuts the circle about <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> in the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. <lb/> |
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1468 Let there be constructed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math>, which extended sill cut the same circle about <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. <lb/> |
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1469 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>p</mi></mstyle></math> be constructed and extended on both sides, which will cut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> in the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <lb/> |
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1470 Then: <lb/> |
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1471 Therefore there is given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>a</mi></mstyle></math>, and the centre of the circle sought. |
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1472 </emph>]<lb/> |
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1473 </s> |
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1474 </p> |
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1475 <pb file="add_6784_f200v" o="200v" n="400"/> |
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1476 <pb file="add_6784_f201" o="201" n="401"/> |
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1477 <div xml:id="echoid-div19" type="page_commentary" level="2" n="19"> |
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1478 <p> |
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1479 <s xml:id="echoid-s81" xml:space="preserve">[<emph style="it">Note: |
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1480 <p> |
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1481 <s xml:id="echoid-s81" xml:space="preserve"> |
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1482 The reference to Pappus is to Commandino's edition of Books III to VIII, |
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1483 <emph style="it">Mathematicae collecitones</emph> (1558). |
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1484 The proposition on page 48v–49 is Proposition IV.15 (not 13, as Harriot appears to have written). |
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1485 A diagram for this proposition appears on Add MS 6784, f. 202; |
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1486 this page shows only calculations of ratios. |
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1487 </s> |
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1488 <lb/> |
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1489 <quote xml:lang="lat"> |
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1490 Theorema XV. Propositio XV. <lb/> |
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1491 Iisdem positis describatur circulus HRT, qui & semicirculos iam dictos, & circulum LGH contingat |
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1492 in punctis HRT, atque a centris A P ad BC basim perpendiculares ducantur AM PN. Dico vt AM vna cum |
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1493 diametro circuli EGH ad diametrum ipsius, ita esse PN ad circuli HRT diametrum. |
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1494 </quote> |
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1495 <lb/> |
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1496 <quote> |
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1497 The same being supposed [as in Proposition 14], there is drawn the circle HRT, which touches both the semicircles |
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1498 already given and the circle LGH, in the points H, R, T. And from the centres A and P to the base there are drawn |
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1499 perpendiculars AM and PN. I say that as AM together with the diameter of the circle EGH is to that that diameter itself, |
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1500 so is PN to the diamter of the circle HRT. |
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1501 </quote> |
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1502 </p> |
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1503 </emph>] |
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1504 <lb/><lb/></s></p></div> |
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1505 <head xml:id="echoid-head69" xml:space="preserve"> |
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1506 5.) pappus. prop. 13. pag. 49. |
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1507 </head> |
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1508 <pb file="add_6784_f201v" o="201v" n="402"/> |
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1509 <pb file="add_6784_f202" o="202" n="403"/> |
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1510 <head xml:id="echoid-head70" xml:space="preserve"> |
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1511 4.) |
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1512 </head> |
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1513 <p> |
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1514 <s xml:id="echoid-s83" xml:space="preserve"> |
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1515 Sint duo circuli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>d</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi><mi>c</mi></mstyle></math> <lb/> |
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1516 contingant se in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <lb/> |
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1517 sit recta per centra <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi><mi>p</mi><mi>c</mi><mi>d</mi></mstyle></math>. <lb/> |
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1518 oportet describere circulum <lb/> |
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1519 contingentem duos circulos <lb/> |
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1520 datos, et lineam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. |
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1521 <lb/>[<emph style="it">tr: |
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1522 Let there be two circles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi><mi>c</mi></mstyle></math> touching in the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <lb/> |
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1523 Let the line through the centre be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi><mi>p</mi><mi>c</mi><mi>d</mi></mstyle></math>. <lb/> |
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1524 One must draw the circle touching the two given circles and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. |
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1525 </emph>]<lb/> |
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1526 </s> |
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1527 </p> |
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1528 <p> |
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1529 <s xml:id="echoid-s84" xml:space="preserve"> |
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1530 <lb/>[...]<lb/> <lb/> |
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1531 Jungantur puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. <lb/> |
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1532 fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mi>k</mi></mstyle></math> parallela <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>d</mi></mstyle></math>. <lb/> |
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1533 <lb/>[...]<lb/> |
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1534 <lb/>[<emph style="it">tr: |
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1535 <lb/>[...]<lb/> <lb/> |
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1536 Let the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> be joined. <lb/> |
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1537 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mi>k</mi></mstyle></math> be parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>d</mi></mstyle></math>. <lb/> |
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1538 <lb/>[...]<lb/> |
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1539 </emph>]<lb/> |
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1540 </s> |
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1541 </p> |
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1542 <p> |
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1543 <s xml:id="echoid-s85" xml:space="preserve"> |
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1544 Bisecetur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>l</mi></mstyle></math>, puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>. <lb/> |
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1545 agatur ad angulos rectos, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>a</mi></mstyle></math>. <lb/> |
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1546 fiat, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>a</mi><mo>=</mo><mi>m</mi><mi>k</mi></mstyle></math>. <lb/> |
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1547 agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>a</mi></mstyle></math>, quæ secabit periferi-<lb/> |
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1548 am minoris circuli in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. <lb/> |
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1549 agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>a</mi><mi>g</mi></mstyle></math>, quæ secabit perife-<lb/> |
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1550 riam maioris circulam in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. <lb/> |
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1551 Dico quod: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>m</mi><mo>=</mo><mi>a</mi><mi>g</mi><mo>=</mo><mi>a</mi><mi>e</mi></mstyle></math>. <lb/> |
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1552 et ideo, circulus per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, <lb/> |
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1553 erit quæsitus. |
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1554 <lb/>[<emph style="it">tr: |
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1555 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>l</mi></mstyle></math> be bisected at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>. <lb/> |
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1556 There is constructed at right angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>a</mi></mstyle></math>. <lb/> |
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1557 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>a</mi><mo>=</mo><mi>m</mi><mi>k</mi></mstyle></math>. <lb/> |
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1558 There is constructed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>a</mi></mstyle></math>, which will cut the circumference of the smaller circle at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. <lb/> |
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1559 There is constructed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>a</mi><mi>g</mi></mstyle></math>, which will cut the circumference of the larger circle at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. <lb/> |
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1560 I say that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>m</mi><mo>=</mo><mi>a</mi><mi>g</mi><mo>=</mo><mi>a</mi><mi>e</mi></mstyle></math>, and therfore the circle through <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> will be the one required. |
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1561 </emph>]<lb/> |
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1562 </s> |
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1563 </p> |
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1564 <pb file="add_6784_f202v" o="202v" n="404"/> |
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1565 <pb file="add_6784_f203" o="203" n="405"/> |
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1566 <div xml:id="echoid-div20" type="page_commentary" level="2" n="20"> |
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1567 <p> |
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1568 <s xml:id="echoid-s86" xml:space="preserve">[<emph style="it">Note: |
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1569 <p> |
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1570 <s xml:id="echoid-s86" xml:space="preserve"> |
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1571 The reference to Pappus is to Commandino's edition of Books III to VIII, |
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1572 <emph style="it">Mathematicae collecitones</emph> (1558). |
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1573 The proposition on page 47 is Proposition IV.14. |
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1574 Harriot's diagram is the same as the one given by Commandino except for his use of lower case letters. |
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1575 A second diagram for the same proposition appears on Add MS 6784, f. 204. |
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1576 </s> |
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1577 <lb/> |
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1578 <quote xml:lang="lat"> |
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1579 Theorema XIIII. Propositio XIIII. <lb/> |
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1580 Sint duo semicirculi BGC BED: & ipsos contingat circulus EFGH: a cuius centro A ad BC basim semicirculorum |
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1581 perpendicularis ducatur AM. Dico ut BM as eam, quæ ex centro circuli EFGH, |
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1582 ita esse in prima figura vtramque simul CB BD ad earum excessum CD; |
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1583 in secunda vero, & tertia figura, ita esse excessum CB BD ad vtramque ipsarum CB BD. |
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1584 </quote> |
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1585 <lb/> |
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1586 <quote> |
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1587 Let there be two semicircles BGC and BED, and their touching circle EFGH, from whose centre A to BC, |
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1588 the base of the semicircle, there is drawn the perpendicular AM. |
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1589 I say that as BM is to that line from the centre of the circle EFGH, |
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1590 inthe first figure will be CB and BD togher to their excess, CD; |
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1591 but in the second and third figure, it will be as the excess of CB over BD to both of CB and BD together. |
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1592 </quote> |
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1593 </p> |
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1594 </emph>] |
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1595 <lb/><lb/></s></p></div> |
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1596 <head xml:id="echoid-head71" xml:space="preserve" xml:lang="lat"> |
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1597 pappus. pag. <lb/> |
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1598 47. |
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1599 <lb/>[<emph style="it">tr: |
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1600 Pappus, page 47. |
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1601 </emph>]<lb/> |
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1602 </head> |
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1603 <pb file="add_6784_f203v" o="203v" n="406"/> |
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1604 <pb file="add_6784_f204" o="204" n="407"/> |
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1605 <div xml:id="echoid-div21" type="page_commentary" level="2" n="21"> |
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1606 <p> |
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1607 <s xml:id="echoid-s88" xml:space="preserve">[<emph style="it">Note: |
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1608 <p> |
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1609 <s xml:id="echoid-s88" xml:space="preserve"> |
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1610 A further diagram for Pappus, <emph style="it">Mathematicae collectiones</emph>, Propostion IV.14. |
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1611 See also the previous folio, Add MS 6784, f. 203. |
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1612 </s> |
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1613 </p> |
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1614 </emph>] |
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1615 <lb/><lb/></s></p></div> |
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1616 <head xml:id="echoid-head72" xml:space="preserve" xml:lang="lat"> |
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1617 pappus. pag. <lb/> |
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1618 47. |
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1619 <lb/>[<emph style="it">tr: |
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1620 Pappus, page 47. |
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1621 </emph>]<lb/> |
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1622 </head> |
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1623 <pb file="add_6784_f204v" o="204v" n="408"/> |
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1624 <pb file="add_6784_f205" o="205" n="409"/> |
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1625 <div xml:id="echoid-div22" type="page_commentary" level="2" n="22"> |
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1626 <p> |
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1627 <s xml:id="echoid-s90" xml:space="preserve">[<emph style="it">Note: |
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1628 <p> |
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1629 <s xml:id="echoid-s90" xml:space="preserve"> |
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1630 Further work on Pappus, Propostion IV.14. |
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1631 </s> |
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1632 </p> |
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1633 </emph>] |
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1634 <lb/><lb/></s></p></div> |
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1635 <head xml:id="echoid-head73" xml:space="preserve"> |
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1636 2) pappus. pag. 47 |
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1637 </head> |
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1638 <pb file="add_6784_f205v" o="205v" n="410"/> |
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1639 <pb file="add_6784_f206" o="206" n="411"/> |
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1640 <div xml:id="echoid-div23" type="page_commentary" level="2" n="23"> |
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1641 <p> |
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1642 <s xml:id="echoid-s92" xml:space="preserve">[<emph style="it">Note: |
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1643 <p> |
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1644 <s xml:id="echoid-s92" xml:space="preserve"> |
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1645 Further work on Pappus, Propostion IV.14. |
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1646 </s> |
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1647 </p> |
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1648 </emph>] |
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1649 <lb/><lb/></s></p></div> |
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1650 <head xml:id="echoid-head74" xml:space="preserve"> |
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1651 3) pappus. <emph style="super">pag.</emph> 47 |
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1652 </head> |
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1653 <pb file="add_6784_f206v" o="206v" n="412"/> |
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1654 <pb file="add_6784_f207" o="207" n="413"/> |
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1655 <div xml:id="echoid-div24" type="page_commentary" level="2" n="24"> |
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1656 <p> |
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1657 <s xml:id="echoid-s94" xml:space="preserve">[<emph style="it">Note: |
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1658 <p> |
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1659 <s xml:id="echoid-s94" xml:space="preserve"> |
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1660 Lists of variations of increasing (c) and decreasing (d) columns, |
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1661 together with other rough work for the 'Magisteria' (Add MS 6782, f. 107 to f. 146v). <lb/> |
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1662 This page is important because it carries a date, day, time, and year: June 28 (Sunday) 10.30am, 1618. |
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1663 </s> |
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1664 </p> |
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1665 </emph>] |
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1666 <lb/><lb/></s></p></div> |
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1667 <p xml:lang="lat"> |
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1668 <s xml:id="echoid-s96" xml:space="preserve"> |
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1669 De causa reflexionis ad angulos æquales. |
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1670 <lb/>[<emph style="it">tr: |
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1671 On the cause of reflection at equal angles. |
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1672 </emph>]<lb/> |
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1673 </s> |
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1674 </p> |
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1675 <p xml:lang="lat"> |
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1676 <s xml:id="echoid-s97" xml:space="preserve"> |
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1677 June 28. .ho: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><mn>1</mn><mn>0</mn><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mstyle></math> <lb/> |
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1678 ante mer: 1618 |
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1679 <lb/>[<emph style="it">tr: |
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|
1680 June 28 (Sunday) 10.30am 1618 |
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1681 </emph>]<lb/> |
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1682 </s> |
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1683 </p> |
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1684 <pb file="add_6784_f207v" o="207v" n="414"/> |
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1685 <div xml:id="echoid-div25" type="page_commentary" level="2" n="25"> |
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1686 <p> |
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1687 <s xml:id="echoid-s98" xml:space="preserve">[<emph style="it">Note: |
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1688 <p> |
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1689 <s xml:id="echoid-s98" xml:space="preserve"> |
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1690 Further lists of variations of increasing (c) and decreasing (d) columns (see Add MS 6784, f. 413). |
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1691 </s> |
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1692 </p> |
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1693 </emph>] |
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1694 <lb/><lb/></s></p></div> |
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1695 <pb file="add_6784_f208" o="208" n="415"/> |
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|
1696 <div xml:id="echoid-div26" type="page_commentary" level="2" n="26"> |
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1697 <p> |
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1698 <s xml:id="echoid-s100" xml:space="preserve">[<emph style="it">Note: |
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1699 <p> |
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1700 <s xml:id="echoid-s100" xml:space="preserve"> |
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1701 Difference tables similar to those on pages 10 and 11 of the 'Magisteria' (Add MS 6782, f. 117 and f. 118). |
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1702 </s> |
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1703 </p> |
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1704 </emph>] |
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1705 <lb/><lb/></s></p></div> |
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1706 <pb file="add_6784_f208v" o="208v" n="416"/> |
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1707 <div xml:id="echoid-div27" type="page_commentary" level="2" n="27"> |
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1708 <p> |
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1709 <s xml:id="echoid-s102" xml:space="preserve">[<emph style="it">Note: |
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1710 <p> |
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1711 <s xml:id="echoid-s102" xml:space="preserve"> |
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|
1712 Formulae for entries in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> column of a difference table, |
|
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|
1713 similar to those on page 14 of the 'Magisteria' (Add MS 6782, f. 121). |
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1714 </s> |
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1715 </p> |
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1716 </emph>] |
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1717 <lb/><lb/></s></p></div> |
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1718 <pb file="add_6784_f209" o="209" n="417"/> |
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|
1719 <pb file="add_6784_f209v" o="209v" n="418"/> |
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1720 <pb file="add_6784_f210" o="210" n="419"/> |
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|
1721 <div xml:id="echoid-div28" type="page_commentary" level="2" n="28"> |
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|
1722 <p> |
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1723 <s xml:id="echoid-s104" xml:space="preserve">[<emph style="it">Note: |
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|
1724 <p> |
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|
1725 <s xml:id="echoid-s104" xml:space="preserve"> |
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|
1726 Rough working for page 15 of the 'Magisteria' (Add MS 6782, f. 122). |
|
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|
1727 </s> |
|
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|
1728 </p> |
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|
1729 </emph>] |
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1730 <lb/><lb/></s></p></div> |
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1731 <pb file="add_6784_f210v" o="210v" n="420"/> |
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1732 <div xml:id="echoid-div29" type="page_commentary" level="2" n="29"> |
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1733 <p> |
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1734 <s xml:id="echoid-s106" xml:space="preserve">[<emph style="it">Note: |
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1735 <p> |
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1736 <s xml:id="echoid-s106" xml:space="preserve"> |
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1737 Formulae for entries in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> columns of a difference table, |
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1738 similar to those on page 14 of the 'Magisteria' (Add MS 6782, f. 121). |
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1739 </s> |
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1740 </p> |
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1741 </emph>] |
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1742 <lb/><lb/></s></p></div> |
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1743 <pb file="add_6784_f211" o="211" n="421"/> |
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1744 <div xml:id="echoid-div30" type="page_commentary" level="2" n="30"> |
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1745 <p> |
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1746 <s xml:id="echoid-s108" xml:space="preserve">[<emph style="it">Note: |
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1747 <p> |
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1748 <s xml:id="echoid-s108" xml:space="preserve"> |
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1749 An incomplete version of the difference table on page 9 of the 'Magisteria' (Add MS 6782, f. 116). |
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1750 </s> |
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1751 </p> |
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1752 </emph>] |
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1753 <lb/><lb/></s></p></div> |
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1754 <pb file="add_6784_f211v" o="211v" n="422"/> |
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1755 <div xml:id="echoid-div31" type="page_commentary" level="2" n="31"> |
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1756 <p> |
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1757 <s xml:id="echoid-s110" xml:space="preserve">[<emph style="it">Note: |
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1758 <p> |
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1759 <s xml:id="echoid-s110" xml:space="preserve"> |
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|
1760 Formulae for entries in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> columns of a difference table; |
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1761 see page 16 of the 'Magisteria' (Add MS 6782, f. 123). |
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1762 </s> |
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1763 </p> |
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1764 </emph>] |
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1765 <lb/><lb/></s></p></div> |
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1766 <pb file="add_6784_f212" o="212" n="423"/> |
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1767 <div xml:id="echoid-div32" type="page_commentary" level="2" n="32"> |
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|
1768 <p> |
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1769 <s xml:id="echoid-s112" xml:space="preserve">[<emph style="it">Note: |
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1770 <p> |
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1771 <s xml:id="echoid-s112" xml:space="preserve"> |
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|
1772 Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). |
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1773 </s> |
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1774 </p> |
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1775 </emph>] |
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1776 <lb/><lb/></s></p></div> |
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1777 <head xml:id="echoid-head75" xml:space="preserve" xml:lang="lat"> |
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1778 Operatio. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. |
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|
1779 <lb/>[<emph style="it">tr: |
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|
1780 Working on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> |
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|
1781 </emph>]<lb/> |
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1782 </head> |
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1783 <p xml:lang="lat"> |
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1784 <s xml:id="echoid-s114" xml:space="preserve"> |
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|
1785 operatio. 1<emph style="super">a</emph> |
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1786 <lb/>[<emph style="it">tr: |
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|
1787 Working (1) |
|
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|
1788 </emph>]<lb/> |
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|
1789 </s> |
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1790 </p> |
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1791 <p xml:lang="lat"> |
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1792 <s xml:id="echoid-s115" xml:space="preserve"> |
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|
1793 operatio. 2<emph style="super">a</emph> |
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1794 <lb/>[<emph style="it">tr: |
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|
1795 Working (2) |
|
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|
1796 </emph>]<lb/> |
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1797 </s> |
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1798 </p> |
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|
1799 <pb file="add_6784_f212v" o="212v" n="424"/> |
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1800 <pb file="add_6784_f213" o="213" n="425"/> |
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1801 <div xml:id="echoid-div33" type="page_commentary" level="2" n="33"> |
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1802 <p> |
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1803 <s xml:id="echoid-s116" xml:space="preserve">[<emph style="it">Note: |
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1804 <p> |
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1805 <s xml:id="echoid-s116" xml:space="preserve"> |
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|
1806 Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). |
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1807 </s> |
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1808 </p> |
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1809 </emph>] |
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1810 <lb/><lb/></s></p></div> |
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1811 <head xml:id="echoid-head76" xml:space="preserve" xml:lang="lat"> |
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1812 Residuum operationis. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. |
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1813 </head> |
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1814 <p xml:lang="lat"> |
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1815 <s xml:id="echoid-s118" xml:space="preserve"> |
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1816 2<emph style="super">a</emph> |
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1817 Working (2) |
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1818 </s> |
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1819 </p> |
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1820 <pb file="add_6784_f213v" o="213v" n="426"/> |
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1821 <pb file="add_6784_f214" o="214" n="427"/> |
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1822 <div xml:id="echoid-div34" type="page_commentary" level="2" n="34"> |
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1823 <p> |
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1824 <s xml:id="echoid-s119" xml:space="preserve">[<emph style="it">Note: |
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1825 <p> |
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1826 <s xml:id="echoid-s119" xml:space="preserve"> |
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1827 General notation for triangular numbers. <lb/> |
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1828 See also page 2 of the 'Magisteria' (Add MS 6782, f. 109). |
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1829 </s> |
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1830 </p> |
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1831 </emph>] |
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1832 <lb/><lb/></s></p></div> |
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1833 <head xml:id="echoid-head77" xml:space="preserve" xml:lang="lat"> |
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1834 3<emph style="super">a</emph> notatio triangularium per notas generales. |
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1835 <lb/>[<emph style="it">tr: |
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1836 3rd notation for triangular numbers, in general symbols. |
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1837 </emph>]<lb/> |
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1838 </head> |
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1839 <pb file="add_6784_f214v" o="214v" n="428"/> |
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1840 <pb file="add_6784_f215" o="215" n="429"/> |
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1841 <pb file="add_6784_f215v" o="215v" n="430"/> |
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1842 <pb file="add_6784_f216" o="216" n="431"/> |
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1843 <div xml:id="echoid-div35" type="page_commentary" level="2" n="35"> |
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1844 <p> |
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1845 <s xml:id="echoid-s121" xml:space="preserve">[<emph style="it">Note: |
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1846 <p> |
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1847 <s xml:id="echoid-s121" xml:space="preserve"> |
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1848 Square roots of binomes of the fifth and sixth kind |
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1849 by the general rule derived in Add MS 6788, f. 15 (and elsewhere). |
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1850 Here Harriot works with two types of fifth binome, |
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1851 (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>c</mi></mrow></msqrt><mo>+</mo><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>d</mi></mrow></msqrt><mo>+</mo><mi>b</mi></mstyle></math>), |
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1852 according to whether the difference between the squares of the two terms is a square or not. |
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1853 Elsewhere he refers to these as bin. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mo>ʹ</mo></mstyle></math> and bin. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mo>ʺ</mo></mstyle></math>. <lb/> |
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1854 Similarly he distinguishes two types of sixth binomes, |
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1855 (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>c</mi><mo>+</mo><mi>d</mi><mi>d</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>b</mi><mi>c</mi></mrow></msqrt><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>c</mi><mo>+</mo><mi>d</mi><mi>f</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>b</mi><mi>c</mi></mrow></msqrt></mstyle></math>). |
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1856 Elsewhere he refers to these as bin. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>ʹ</mo></mstyle></math> and bin. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>ʺ</mo></mstyle></math>. <lb/> |
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1857 In all cases the roots are cross-checked. |
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1858 </s> |
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1859 </p> |
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1860 </emph>] |
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1861 <lb/><lb/></s></p></div> |
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1862 <pb file="add_6784_f216v" o="216v" n="432"/> |
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1863 <pb file="add_6784_f217" o="217" n="433"/> |
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1864 <div xml:id="echoid-div36" type="page_commentary" level="2" n="36"> |
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1865 <p> |
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1866 <s xml:id="echoid-s123" xml:space="preserve">[<emph style="it">Note: |
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1867 <p> |
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1868 <s xml:id="echoid-s123" xml:space="preserve"> |
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1869 Square roots of binomes of the third and fourth kind |
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1870 (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mi>c</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mi>c</mi><mo>-</mo><mi>d</mi><mi>d</mi><mi>c</mi></mrow></msqrt></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mo>-</mo><mi>b</mi><mi>d</mi></mrow></msqrt></mstyle></math>), |
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1871 by the general rule derived in Add MS 6788, f. 15 (and elsewhere). |
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1872 In both cases the roots are checked by multiplication. |
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1873 </s> |
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1874 </p> |
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1875 </emph>] |
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1876 <lb/><lb/></s></p></div> |
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1877 <pb file="add_6784_f217v" o="217v" n="434"/> |
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1878 <pb file="add_6784_f218" o="218" n="435"/> |
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1879 <div xml:id="echoid-div37" type="page_commentary" level="2" n="37"> |
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1880 <p> |
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1881 <s xml:id="echoid-s125" xml:space="preserve">[<emph style="it">Note: |
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1882 <p> |
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1883 <s xml:id="echoid-s125" xml:space="preserve"> |
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1884 Square roots of binomes of the first and second kind |
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1885 (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>c</mi></mrow></msqrt></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mo maxsize="1">(</mo><mi>b</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>d</mi><mo maxsize="1">)</mo></mrow></msqrt><mo>+</mo><mi>b</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>d</mi></mstyle></math>), |
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1886 by the general rule derived in Add MS 6788, f. 15 (and elsewhere). |
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1887 In both cases the roots are checked by multiplication. |
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1888 </s> |
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1889 </p> |
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1890 </emph>] |
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1891 <lb/><lb/></s></p></div> |
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1892 <pb file="add_6784_f218v" o="218v" n="436"/> |
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1893 <pb file="add_6784_f219" o="219" n="437"/> |
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1894 <div xml:id="echoid-div38" type="page_commentary" level="2" n="38"> |
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1895 <p> |
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1896 <s xml:id="echoid-s127" xml:space="preserve">[<emph style="it">Note: |
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1897 <p> |
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1898 <s xml:id="echoid-s127" xml:space="preserve"> |
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1899 Square roots of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mi>b</mi><mi>b</mi><mi>d</mi><mi>d</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mi>d</mi><mi>d</mi><mo>-</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>d</mi></mrow></msqrt></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mi>c</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>d</mi><mi>d</mi><mi>c</mi></mrow></msqrt></mstyle></math>, |
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1900 by the general rule derived in Add MS 6788, f. 15 (and elsewhere). In each case, the root is checked by multiplication. |
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1901 The numerical examples in Add MS 6783, f. 360v, f. 361, and Add MS 6782, f. 228, |
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1902 are closely related to the work on this page. |
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1903 </s> |
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1904 </p> |
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1905 </emph>] |
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1906 <lb/><lb/></s></p></div> |
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1907 <p xml:lang="lat"> |
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1908 <s xml:id="echoid-s129" xml:space="preserve"> |
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1909 Nam: eius quadratum |
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1910 <lb/>[<emph style="it">tr: |
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1911 For: its square |
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1912 </emph>]<lb/> |
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1913 </s> |
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1914 </p> |
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1915 <p xml:lang="lat"> |
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1916 <s xml:id="echoid-s130" xml:space="preserve"> |
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1917 Quia: duo quad: |
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1918 <lb/>[<emph style="it">tr: |
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1919 Because: two squares |
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1920 </emph>]<lb/> |
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1921 </s> |
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1922 </p> |
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1923 <p xml:lang="lat"> |
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1924 <s xml:id="echoid-s131" xml:space="preserve"> |
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1925 Et: duo rectang: |
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1926 <lb/>[<emph style="it">tr: |
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1927 And: two rectangles |
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1928 </emph>]<lb/> |
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1929 </s> |
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1930 </p> |
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1931 <pb file="add_6784_f219v" o="219v" n="438"/> |
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1932 <pb file="add_6784_f220" o="220" n="439"/> |
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1933 <div xml:id="echoid-div39" type="page_commentary" level="2" n="39"> |
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1934 <p> |
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1935 <s xml:id="echoid-s132" xml:space="preserve">[<emph style="it">Note: |
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1936 <p> |
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1937 <s xml:id="echoid-s132" xml:space="preserve"> |
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1938 Square roots of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>c</mi></mrow></msqrt><mo>+</mo><mn>2</mn><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>d</mi></mrow></msqrt><mo>+</mo><mn>2</mn><mi>b</mi><mi>b</mi><mi>b</mi><mi>d</mi><mo>-</mo><mn>2</mn><mi>b</mi><mi>d</mi><mi>d</mi><mi>d</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>c</mi></mrow></msqrt><mo>+</mo><mn>4</mn><mi>d</mi><mi>c</mi></mstyle></math>, |
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1939 by the general rule derived in Add MS 6788, f. 15 (and elsewhere). In each case, the root is checked by multiplication. |
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1940 </s> |
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1941 </p> |
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1942 </emph>] |
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1943 <lb/><lb/></s></p></div> |
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1944 <pb file="add_6784_f220v" o="220v" n="440"/> |
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1945 <pb file="add_6784_f221" o="221" n="441"/> |
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1946 <head xml:id="echoid-head78" xml:space="preserve" xml:lang="lat"> |
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1947 Examinatio æquationis per numeros |
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1948 <lb/>[<emph style="it">tr: |
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1949 An examination of an equation in numbers |
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1950 </emph>]<lb/> |
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1951 </head> |
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1952 <p xml:lang="lat"> |
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1953 <s xml:id="echoid-s134" xml:space="preserve"> |
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1954 et ita est (ut supra) |
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1955 <lb/>[<emph style="it">tr: |
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1956 and so it is (as above) |
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1957 </emph>]<lb/> |
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1958 </s> |
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1959 </p> |
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1960 <p xml:lang="lat"> |
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1961 <s xml:id="echoid-s135" xml:space="preserve"> |
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1962 et pro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> |
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1963 <lb/>[<emph style="it">tr: |
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1964 and for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> |
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1965 </emph>]<lb/> |
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1966 </s> |
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1967 <lb/> |
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1968 <s xml:id="echoid-s136" xml:space="preserve"> |
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1969 et ita est (ut infra) |
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1970 <lb/>[<emph style="it">tr: |
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1971 and so it is (as below) |
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1972 </emph>]<lb/> |
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1973 </s> |
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1974 </p> |
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1975 <pb file="add_6784_f221v" o="221v" n="442"/> |
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1976 <pb file="add_6784_f222" o="222" n="443"/> |
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1977 <pb file="add_6784_f222v" o="222v" n="444"/> |
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1978 <pb file="add_6784_f223" o="223" n="445"/> |
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1979 <pb file="add_6784_f223v" o="223v" n="446"/> |
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1980 <pb file="add_6784_f224" o="224" n="447"/> |
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1981 <pb file="add_6784_f224v" o="224v" n="448"/> |
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1982 <pb file="add_6784_f225" o="225" n="449"/> |
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1983 <pb file="add_6784_f225v" o="225v" n="450"/> |
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1984 <pb file="add_6784_f226" o="226" n="451"/> |
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1985 <pb file="add_6784_f226v" o="226v" n="452"/> |
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1986 <pb file="add_6784_f227" o="227" n="453"/> |
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1987 <pb file="add_6784_f227v" o="227v" n="454"/> |
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1988 <pb file="add_6784_f228" o="228" n="455"/> |
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1989 <pb file="add_6784_f228v" o="228v" n="456"/> |
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1990 <pb file="add_6784_f229" o="229" n="457"/> |
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1991 <pb file="add_6784_f229v" o="229v" n="458"/> |
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1992 <pb file="add_6784_f230" o="230" n="459"/> |
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1993 <pb file="add_6784_f230v" o="230v" n="460"/> |
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1994 <pb file="add_6784_f231" o="231" n="461"/> |
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1995 <pb file="add_6784_f231v" o="231v" n="462"/> |
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1996 <pb file="add_6784_f232" o="232" n="463"/> |
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1997 <pb file="add_6784_f232v" o="232v" n="464"/> |
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1998 <pb file="add_6784_f233" o="233" n="465"/> |
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1999 <pb file="add_6784_f233v" o="233v" n="466"/> |
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2000 <pb file="add_6784_f234" o="234" n="467"/> |
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2001 <pb file="add_6784_f234v" o="234v" n="468"/> |
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2002 <pb file="add_6784_f235" o="235" n="469"/> |
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2003 <pb file="add_6784_f235v" o="235v" n="470"/> |
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2004 <pb file="add_6784_f236" o="236" n="471"/> |
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2005 <pb file="add_6784_f236v" o="236v" n="472"/> |
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2006 <pb file="add_6784_f237" o="237" n="473"/> |
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2007 <pb file="add_6784_f237v" o="237v" n="474"/> |
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2008 <pb file="add_6784_f238" o="238" n="475"/> |
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2009 <pb file="add_6784_f238v" o="238v" n="476"/> |
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2010 <pb file="add_6784_f239" o="239" n="477"/> |
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2011 <pb file="add_6784_f239v" o="239v" n="478"/> |
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2012 <pb file="add_6784_f240" o="240" n="479"/> |
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2013 <pb file="add_6784_f240v" o="240v" n="480"/> |
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2014 <pb file="add_6784_f241" o="241" n="481"/> |
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2015 <pb file="add_6784_f241v" o="241v" n="482"/> |
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2016 <pb file="add_6784_f242" o="242" n="483"/> |
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2017 <pb file="add_6784_f242v" o="242v" n="484"/> |
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2018 <pb file="add_6784_f243" o="243" n="485"/> |
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2019 <pb file="add_6784_f243v" o="243v" n="486"/> |
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2020 <pb file="add_6784_f244" o="244" n="487"/> |
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2021 <pb file="add_6784_f244v" o="244v" n="488"/> |
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2022 <pb file="add_6784_f245" o="245" n="489"/> |
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2023 <pb file="add_6784_f245v" o="245v" n="490"/> |
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2024 <pb file="add_6784_f246" o="246" n="491"/> |
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2025 <head xml:id="echoid-head79" xml:space="preserve" xml:lang="lat"> |
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2026 3. |
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2027 </head> |
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2028 <p xml:lang="lat"> |
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2029 <s xml:id="echoid-s137" xml:space="preserve"> |
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2030 In Achille |
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2031 <lb/>[<emph style="it">tr: |
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2032 On Achilles |
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2033 </emph>]<lb/> |
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2034 </s> |
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2035 <lb/> |
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2036 <s xml:id="echoid-s138" xml:space="preserve"> |
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2037 vel per æquationem rationum |
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2038 <lb/>[<emph style="it">tr: |
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2039 or by the equality of ratios |
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2040 </emph>]<lb/> |
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2041 </s> |
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2042 </p> |
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2043 <p xml:lang="lat"> |
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2044 <s xml:id="echoid-s139" xml:space="preserve"> |
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2045 Aliter |
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2046 <lb/>[<emph style="it">tr: |
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2047 Another way |
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2048 </emph>]<lb/> |
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2049 </s> |
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2050 <lb/> |
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2051 <s xml:id="echoid-s140" xml:space="preserve"> |
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2052 Sit ratio motus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mo>=</mo><mi>c</mi><mi>o</mi></mstyle></math> |
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2053 <lb/>[<emph style="it">tr: |
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2054 Let the ratio of motion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>o</mi></mstyle></math>. |
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|
2055 </emph>]<lb/> |
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2056 </s> |
|
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|
2057 <lb/> |
|
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2058 <s xml:id="echoid-s141" xml:space="preserve"> |
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|
2059 Tempus Tempus |
|
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2060 <lb/>[<emph style="it">tr: |
|
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|
2061 Time; Time |
|
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|
2062 </emph>]<lb/> |
|
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|
2063 </s> |
|
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|
2064 </p> |
|
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|
2065 <p xml:lang="lat"> |
|
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|
2066 <s xml:id="echoid-s142" xml:space="preserve"> |
|
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|
2067 Aliter |
|
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|
2068 <lb/>[<emph style="it">tr: |
|
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|
2069 Another way |
|
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|
2070 </emph>]<lb/> |
|
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|
2071 </s> |
|
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|
2072 </p> |
|
22d6a63640c6
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|
2073 <pb file="add_6784_f246v" o="246v" n="492"/> |
|
22d6a63640c6
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|
2074 <pb file="add_6784_f247" o="247" n="493"/> |
|
22d6a63640c6
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|
2075 <head xml:id="echoid-head80" xml:space="preserve" xml:lang="lat"> |
|
22d6a63640c6
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|
2076 4. |
|
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|
2077 </head> |
|
22d6a63640c6
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|
2078 <pb file="add_6784_f247v" o="247v" n="494"/> |
|
22d6a63640c6
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|
2079 <pb file="add_6784_f248" o="248" n="495"/> |
|
22d6a63640c6
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|
2080 <head xml:id="echoid-head81" xml:space="preserve" xml:lang="lat"> |
|
22d6a63640c6
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|
2081 5. |
|
22d6a63640c6
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|
2082 </head> |
|
22d6a63640c6
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|
2083 <pb file="add_6784_f248v" o="248v" n="496"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2084 <pb file="add_6784_f249" o="249" n="497"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2085 <pb file="add_6784_f249v" o="249v" n="498"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2086 <pb file="add_6784_f250" o="250" n="499"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2087 <pb file="add_6784_f250v" o="250v" n="500"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2088 <pb file="add_6784_f251" o="251" n="501"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2089 <pb file="add_6784_f251v" o="251v" n="502"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2090 <pb file="add_6784_f252" o="252" n="503"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2091 <pb file="add_6784_f252v" o="252v" n="504"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2092 <pb file="add_6784_f253" o="253" n="505"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2093 <pb file="add_6784_f253v" o="253v" n="506"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2094 <pb file="add_6784_f254" o="254" n="507"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2095 <pb file="add_6784_f254v" o="254v" n="508"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2096 <pb file="add_6784_f255" o="255" n="509"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2097 <pb file="add_6784_f255v" o="255v" n="510"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2098 <pb file="add_6784_f256" o="256" n="511"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2099 <pb file="add_6784_f256v" o="256v" n="512"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2100 <pb file="add_6784_f257" o="257" n="513"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2101 <pb file="add_6784_f257v" o="257v" n="514"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2102 <pb file="add_6784_f258" o="258" n="515"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2103 <pb file="add_6784_f258v" o="258v" n="516"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2104 <pb file="add_6784_f259" o="259" n="517"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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|
2105 <pb file="add_6784_f259v" o="259v" n="518"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2106 <pb file="add_6784_f260" o="260" n="519"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2107 <pb file="add_6784_f260v" o="260v" n="520"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2108 <pb file="add_6784_f261" o="261" n="521"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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|
2109 <pb file="add_6784_f261v" o="261v" n="522"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2110 <pb file="add_6784_f262" o="262" n="523"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2111 <pb file="add_6784_f262v" o="262v" n="524"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2112 <pb file="add_6784_f263" o="263" n="525"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2113 <pb file="add_6784_f263v" o="263v" n="526"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2114 <pb file="add_6784_f264" o="264" n="527"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2115 <pb file="add_6784_f264v" o="264v" n="528"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2116 <pb file="add_6784_f265" o="265" n="529"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2117 <pb file="add_6784_f265v" o="265v" n="530"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2118 <pb file="add_6784_f266" o="266" n="531"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2119 <pb file="add_6784_f266v" o="266v" n="532"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2120 <pb file="add_6784_f267" o="267" n="533"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2121 <pb file="add_6784_f267v" o="267v" n="534"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2122 <pb file="add_6784_f268" o="268" n="535"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2123 <pb file="add_6784_f268v" o="268v" n="536"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2124 <pb file="add_6784_f269" o="269" n="537"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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|
2125 <pb file="add_6784_f269v" o="269v" n="538"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2126 <pb file="add_6784_f270" o="270" n="539"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2127 <pb file="add_6784_f270v" o="270v" n="540"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2128 <pb file="add_6784_f271" o="271" n="541"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2129 <pb file="add_6784_f271v" o="271v" n="542"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2130 <pb file="add_6784_f272" o="272" n="543"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2131 <pb file="add_6784_f272v" o="272v" n="544"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2132 <pb file="add_6784_f273" o="273" n="545"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2133 <pb file="add_6784_f273v" o="273v" n="546"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2134 <pb file="add_6784_f274" o="274" n="547"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2135 <pb file="add_6784_f274v" o="274v" n="548"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
2136 <pb file="add_6784_f275" o="275" n="549"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
2137 <pb file="add_6784_f275v" o="275v" n="550"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2138 <pb file="add_6784_f276" o="276" n="551"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
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|
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|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
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|
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|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
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|
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|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
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|
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|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
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|
2143 <pb file="add_6784_f278v" o="278v" n="556"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
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|
2144 <pb file="add_6784_f279" o="279" n="557"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
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|
2145 <pb file="add_6784_f279v" o="279v" n="558"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
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|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
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|
2147 <pb file="add_6784_f280v" o="280v" n="560"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2148 <pb file="add_6784_f281" o="281" n="561"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
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|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2150 <pb file="add_6784_f282" o="282" n="563"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2151 <pb file="add_6784_f282v" o="282v" n="564"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2152 <pb file="add_6784_f283" o="283" n="565"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2153 <pb file="add_6784_f283v" o="283v" n="566"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2154 <pb file="add_6784_f284" o="284" n="567"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2155 <pb file="add_6784_f284v" o="284v" n="568"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2156 <pb file="add_6784_f285" o="285" n="569"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2157 <pb file="add_6784_f285v" o="285v" n="570"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2158 <pb file="add_6784_f286" o="286" n="571"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2159 <pb file="add_6784_f286v" o="286v" n="572"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2160 <pb file="add_6784_f287" o="287" n="573"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2161 <pb file="add_6784_f287v" o="287v" n="574"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2162 <pb file="add_6784_f288" o="288" n="575"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2163 <pb file="add_6784_f288v" o="288v" n="576"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2164 <pb file="add_6784_f289" o="289" n="577"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2165 <pb file="add_6784_f289v" o="289v" n="578"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2166 <pb file="add_6784_f290" o="290" n="579"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2167 <pb file="add_6784_f290v" o="290v" n="580"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2168 <pb file="add_6784_f291" o="291" n="581"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2169 <pb file="add_6784_f291v" o="291v" n="582"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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changeset
|
2170 <pb file="add_6784_f292" o="292" n="583"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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changeset
|
2171 <pb file="add_6784_f292v" o="292v" n="584"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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changeset
|
2172 <pb file="add_6784_f293" o="293" n="585"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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changeset
|
2173 <pb file="add_6784_f293v" o="293v" n="586"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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changeset
|
2174 <pb file="add_6784_f294" o="294" n="587"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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changeset
|
2175 <pb file="add_6784_f294v" o="294v" n="588"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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changeset
|
2176 <pb file="add_6784_f295" o="295" n="589"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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changeset
|
2177 <pb file="add_6784_f295v" o="295v" n="590"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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changeset
|
2178 <pb file="add_6784_f296" o="296" n="591"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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changeset
|
2179 <pb file="add_6784_f296v" o="296v" n="592"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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changeset
|
2180 <pb file="add_6784_f297" o="297" n="593"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2181 <pb file="add_6784_f297v" o="297v" n="594"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2182 <pb file="add_6784_f298" o="298" n="595"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2183 <pb file="add_6784_f298v" o="298v" n="596"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2184 <pb file="add_6784_f299" o="299" n="597"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2185 <pb file="add_6784_f299v" o="299v" n="598"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2186 <pb file="add_6784_f300" o="300" n="599"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2187 <pb file="add_6784_f300v" o="300v" n="600"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2188 <pb file="add_6784_f301" o="301" n="601"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2189 <pb file="add_6784_f301v" o="301v" n="602"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2190 <pb file="add_6784_f302" o="302" n="603"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2191 <pb file="add_6784_f302v" o="302v" n="604"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
2192 <pb file="add_6784_f303" o="303" n="605"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2193 <pb file="add_6784_f303v" o="303v" n="606"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2194 <pb file="add_6784_f304" o="304" n="607"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2195 <pb file="add_6784_f304v" o="304v" n="608"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2196 <pb file="add_6784_f305" o="305" n="609"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2197 <pb file="add_6784_f305v" o="305v" n="610"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2198 <pb file="add_6784_f306" o="306" n="611"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2199 <pb file="add_6784_f306v" o="306v" n="612"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2200 <pb file="add_6784_f307" o="307" n="613"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2201 <pb file="add_6784_f307v" o="307v" n="614"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2202 <pb file="add_6784_f308" o="308" n="615"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2203 <pb file="add_6784_f308v" o="308v" n="616"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2204 <pb file="add_6784_f309" o="309" n="617"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2205 <pb file="add_6784_f309v" o="309v" n="618"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2206 <pb file="add_6784_f310" o="310" n="619"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
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changeset
|
2207 <pb file="add_6784_f310v" o="310v" n="620"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2208 <pb file="add_6784_f311" o="311" n="621"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2209 <pb file="add_6784_f311v" o="311v" n="622"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2210 <pb file="add_6784_f312" o="312" n="623"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2211 <pb file="add_6784_f312v" o="312v" n="624"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
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|
2212 <pb file="add_6784_f313" o="313" n="625"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2213 <pb file="add_6784_f313v" o="313v" n="626"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2214 <pb file="add_6784_f314" o="314" n="627"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
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|
2215 <pb file="add_6784_f314v" o="314v" n="628"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
2216 <pb file="add_6784_f315" o="315" n="629"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
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|
2217 <pb file="add_6784_f315v" o="315v" n="630"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
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|
2218 <pb file="add_6784_f316" o="316" n="631"/> |
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2219 <pb file="add_6784_f316v" o="316v" n="632"/> |
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2220 <pb file="add_6784_f317" o="317" n="633"/> |
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2221 <pb file="add_6784_f317v" o="317v" n="634"/> |
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2222 <pb file="add_6784_f318" o="318" n="635"/> |
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2223 <pb file="add_6784_f318v" o="318v" n="636"/> |
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2224 <pb file="add_6784_f319" o="319" n="637"/> |
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2225 <pb file="add_6784_f319v" o="319v" n="638"/> |
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2226 <pb file="add_6784_f320" o="320" n="639"/> |
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2227 <pb file="add_6784_f320v" o="320v" n="640"/> |
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2228 <pb file="add_6784_f321" o="321" n="641"/> |
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2229 <pb file="add_6784_f321v" o="321v" n="642"/> |
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2230 <div xml:id="echoid-div40" type="page_commentary" level="2" n="40"> |
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2231 <p> |
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2232 <s xml:id="echoid-s143" xml:space="preserve">[<emph style="it">Note: |
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2233 <p> |
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2234 <s xml:id="echoid-s143" xml:space="preserve"> |
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2235 The verses on this page describe the rules for operating with |
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2236 positive quantities ('more') and negative quantites ('lesse'). |
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2237 The first verse sets out the rules for multiplication. |
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2238 The second and third verses deal with subtraction of a negative quantity from a negative quantity, |
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2239 where the result may be either positive or negative. <lb/> |
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2240 Like folios Add MS 6784, f. 323, f. 324, which follow soon after it, this one appears to be based on Viète, |
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2241 <emph style="it">In artem analyticen isagoge</emph>, 1591, |
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2242 in this case on Chapter IV, Praeceptum II and Praeceptum III. |
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2243 </s> |
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2244 </p> |
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2245 </emph>] |
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2246 <lb/><lb/></s></p></div> |
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2247 <p> |
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2248 <s xml:id="echoid-s145" xml:space="preserve"> |
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2249 If more by more must needes make more <lb/> |
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2250 Then lesse by more makes lesse of more <lb/> |
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2251 And lesse by lesse makes lesse of lesse <lb/> |
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2252 If more be more and lesse be lesse. |
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2253 </s> |
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2254 </p> |
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2255 <p> |
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2256 <s xml:id="echoid-s146" xml:space="preserve"> |
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2257 Yet lesse of lesse makes lesse or more <lb/> |
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2258 The which is best keep both in store <lb/> |
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2259 If lesse of lesse thou <emph style="super">you</emph> wilt make lesse <lb/> |
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2260 Then pull <emph style="super">bate</emph> the same from that is lesse |
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2261 </s> |
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2262 </p> |
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2263 <p> |
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2264 <s xml:id="echoid-s147" xml:space="preserve"> |
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2265 But if the same thou <emph style="super">you</emph> wilt make more <lb/> |
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2266 Then add the same <emph style="super">to it</emph> to that is <emph style="super">the sign of</emph> more <lb/> |
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2267 The signe <emph style="super">rule</emph> of more is best to use <lb/> |
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2268 Except some <emph style="super">Yet for some</emph> cause |
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2269 the <emph style="super">do</emph> other choose <emph style="super">then it refuse</emph> <lb/> |
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2270 For <emph style="super">So</emph> <emph style="super">Yet</emph> both are one, for both are true <lb/> |
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2271 of this inough and so adew. |
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2272 </s> |
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2273 </p> |
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2274 <pb file="add_6784_f322" o="322" n="643"/> |
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2275 <div xml:id="echoid-div41" type="page_commentary" level="2" n="41"> |
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2276 <p> |
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2277 <s xml:id="echoid-s148" xml:space="preserve">[<emph style="it">Note: |
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2278 <p> |
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2279 <s xml:id="echoid-s148" xml:space="preserve"> |
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2280 This page shows several examples of additions and subtractions using letters. |
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2281 Note that here such operations are only carried out between quantities of the same dimension. |
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2282 </s> |
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2283 </p> |
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2284 </emph>] |
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2285 <lb/><lb/></s></p></div> |
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2286 <head xml:id="echoid-head82" xml:space="preserve" xml:lang="lat"> |
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2287 1) Operationes logisticæ, in notis |
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2288 <lb/>[<emph style="it">tr: |
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2289 The operations of arithmetic in symbols. |
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2290 </emph>]<lb/> |
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2291 </head> |
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2292 <p xml:lang="lat"> |
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2293 <s xml:id="echoid-s150" xml:space="preserve"> |
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2294 adde |
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2295 <lb/>[<emph style="it">tr: |
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2296 add |
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2297 </emph>]<lb/> |
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2298 </s> |
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2299 <lb/> |
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2300 <s xml:id="echoid-s151" xml:space="preserve"> |
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2301 summa |
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2302 <lb/>[<emph style="it">tr: |
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2303 sum |
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2304 </emph>]<lb/> |
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2305 </s> |
|
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|
2306 </p> |
|
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|
2307 <p xml:lang="lat"> |
|
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2308 <s xml:id="echoid-s152" xml:space="preserve"> |
|
22d6a63640c6
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|
2309 subduce |
|
22d6a63640c6
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|
2310 <lb/>[<emph style="it">tr: |
|
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|
2311 subtract |
|
22d6a63640c6
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|
2312 </emph>]<lb/> |
|
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|
2313 </s> |
|
22d6a63640c6
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|
2314 <lb/> |
|
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2315 <s xml:id="echoid-s153" xml:space="preserve"> |
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|
2316 reliqua |
|
22d6a63640c6
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|
2317 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
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|
2318 remainder |
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22d6a63640c6
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|
2319 </emph>]<lb/> |
|
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|
2320 </s> |
|
22d6a63640c6
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|
2321 </p> |
|
22d6a63640c6
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|
2322 <pb file="add_6784_f322v" o="322v" n="644"/> |
|
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|
2323 <pb file="add_6784_f323" o="323" n="645"/> |
|
22d6a63640c6
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|
2324 <div xml:id="echoid-div42" type="page_commentary" level="2" n="42"> |
|
22d6a63640c6
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|
2325 <p> |
|
22d6a63640c6
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|
2326 <s xml:id="echoid-s154" xml:space="preserve">[<emph style="it">Note: |
|
22d6a63640c6
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|
2327 <p> |
|
22d6a63640c6
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|
2328 <s xml:id="echoid-s154" xml:space="preserve"> |
|
22d6a63640c6
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|
2329 This page shows examples of multiplication and division using letters. |
|
22d6a63640c6
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|
2330 </s> |
|
22d6a63640c6
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|
2331 </p> |
|
22d6a63640c6
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|
2332 </emph>] |
|
22d6a63640c6
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|
2333 <lb/><lb/></s></p></div> |
|
22d6a63640c6
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|
2334 <head xml:id="echoid-head83" xml:space="preserve"> |
|
22d6a63640c6
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|
2335 2) |
|
22d6a63640c6
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|
2336 </head> |
|
22d6a63640c6
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|
2337 <p xml:lang="lat"> |
|
22d6a63640c6
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|
2338 <s xml:id="echoid-s156" xml:space="preserve"> |
|
22d6a63640c6
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|
2339 multip. |
|
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|
2340 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
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|
2341 multiply |
|
22d6a63640c6
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|
2342 </emph>]<lb/> |
|
22d6a63640c6
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|
2343 </s> |
|
22d6a63640c6
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|
2344 <lb/> |
|
22d6a63640c6
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|
2345 <s xml:id="echoid-s157" xml:space="preserve"> |
|
22d6a63640c6
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|
2346 in |
|
22d6a63640c6
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|
2347 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
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|
2348 by |
|
22d6a63640c6
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|
2349 </emph>]<lb/> |
|
22d6a63640c6
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|
2350 </s> |
|
22d6a63640c6
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|
2351 <lb/> |
|
22d6a63640c6
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|
2352 <s xml:id="echoid-s158" xml:space="preserve"> |
|
22d6a63640c6
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|
2353 facta |
|
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|
2354 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
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|
2355 product |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2356 </emph>]<lb/> |
|
22d6a63640c6
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|
2357 </s> |
|
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|
2358 </p> |
|
22d6a63640c6
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|
2359 <p xml:lang="lat"> |
|
22d6a63640c6
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|
2360 <s xml:id="echoid-s159" xml:space="preserve"> |
|
22d6a63640c6
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|
2361 applica |
|
22d6a63640c6
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|
2362 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
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|
2363 divide |
|
22d6a63640c6
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|
2364 </emph>]<lb/> |
|
22d6a63640c6
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|
2365 </s> |
|
22d6a63640c6
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|
2366 <lb/> |
|
22d6a63640c6
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|
2367 <s xml:id="echoid-s160" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2368 ad |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2369 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2370 by |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2371 </emph>]<lb/> |
|
22d6a63640c6
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|
2372 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2373 <lb/> |
|
22d6a63640c6
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|
2374 <s xml:id="echoid-s161" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
2375 orta |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2376 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
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changeset
|
2377 result |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2378 </emph>]<lb/> |
|
22d6a63640c6
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|
2379 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2380 </p> |
|
22d6a63640c6
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changeset
|
2381 <p xml:lang="lat"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
2382 <s xml:id="echoid-s162" xml:space="preserve"> |
|
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
|
2383 manifestum <lb/> |
|
22d6a63640c6
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changeset
|
2384 per præcog-<lb/> |
|
22d6a63640c6
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changeset
|
2385 nitam genera-<lb/> |
|
22d6a63640c6
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|
2386 tionem. |
|
22d6a63640c6
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|
2387 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
2388 evident from the previously learned constructions |
|
22d6a63640c6
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|
2389 </emph>]<lb/> |
|
22d6a63640c6
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changeset
|
2390 </s> |
|
22d6a63640c6
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changeset
|
2391 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
2392 <pb file="add_6784_f323v" o="323v" n="646"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
2393 <pb file="add_6784_f324" o="324" n="647"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
2394 <div xml:id="echoid-div43" type="page_commentary" level="2" n="43"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
2395 <p> |
|
22d6a63640c6
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changeset
|
2396 <s xml:id="echoid-s163" xml:space="preserve">[<emph style="it">Note: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2397 <p> |
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2398 <s xml:id="echoid-s163" xml:space="preserve"> |
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2399 The examples of division on this page are taken directly from Viète, |
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2400 <emph style="it">In artem analyticen isagoge</emph>, 1591, Chapter IV, end of Praeceptum IV, |
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2401 but Harriot has re-written the examples in his own symbolic notation. |
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2402 </s> |
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2403 </p> |
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2404 </emph>] |
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2405 <lb/><lb/></s></p></div> |
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2406 <head xml:id="echoid-head84" xml:space="preserve"> |
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2407 3) |
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2408 </head> |
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2409 <pb file="add_6784_f324v" o="324v" n="648"/> |
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2410 <pb file="add_6784_f325" o="325" n="649"/> |
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2411 <head xml:id="echoid-head85" xml:space="preserve"> |
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2412 4) |
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2413 </head> |
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2414 <div xml:id="echoid-div44" type="page_commentary" level="2" n="44"> |
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2415 <p> |
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2416 <s xml:id="echoid-s165" xml:space="preserve">[<emph style="it">Note: |
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2417 <p> |
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2418 <s xml:id="echoid-s165" xml:space="preserve"> |
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2419 The terminology and examples on this page are taken directly from Viète, |
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2420 <emph style="it">In artem analyticen isagoge</emph>, 1591, Chapter V, |
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2421 but Harriot has re-written the examples in his own symbolic notation. |
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2422 </s> |
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2423 </p> |
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2424 </emph>] |
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2425 <lb/><lb/></s></p></div> |
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2426 <p xml:lang=""> |
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2427 <s xml:id="echoid-s167" xml:space="preserve"> |
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2428 Sit: |
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2429 <lb/>[<emph style="it">tr: |
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2430 Let: |
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2431 </emph>]<lb/> |
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2432 </s> |
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2433 <lb/> |
|
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2434 <s xml:id="echoid-s168" xml:space="preserve"> |
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|
2435 Dico quod: per Antithesin. |
|
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2436 <lb/>[<emph style="it">tr: |
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|
2437 I say that, by antihesis: |
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2438 </emph>]<lb/> |
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2439 </s> |
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2440 <lb/> |
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2441 <s xml:id="echoid-s169" xml:space="preserve"> |
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|
2442 Quoniam: |
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2443 <lb/>[<emph style="it">tr: |
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|
2444 Because: |
|
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2445 </emph>]<lb/> |
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2446 </s> |
|
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2447 <lb/> |
|
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2448 <s xml:id="echoid-s170" xml:space="preserve"> |
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|
2449 Adde utrolique. |
|
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|
2450 <lb/>[<emph style="it">tr: |
|
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|
2451 Add to each side. |
|
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|
2452 </emph>]<lb/> |
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|
2453 </s> |
|
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2454 <lb/> |
|
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|
2455 <s xml:id="echoid-s171" xml:space="preserve"> |
|
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|
2456 Ergo: |
|
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|
2457 <lb/>[<emph style="it">tr: |
|
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|
2458 Therefore: |
|
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|
2459 </emph>]<lb/> |
|
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|
2460 </s> |
|
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|
2461 </p> |
|
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|
2462 <p xml:lang=""> |
|
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|
2463 <s xml:id="echoid-s172" xml:space="preserve"> |
|
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|
2464 Secundo: sit, |
|
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|
2465 <lb/>[<emph style="it">tr: |
|
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|
2466 Second, let: |
|
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|
2467 </emph>]<lb/> |
|
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|
2468 </s> |
|
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|
2469 <lb/> |
|
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|
2470 <s xml:id="echoid-s173" xml:space="preserve"> |
|
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|
2471 Dico quod: |
|
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|
2472 <lb/>[<emph style="it">tr: |
|
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|
2473 I say that: |
|
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|
2474 </emph>]<lb/> |
|
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|
2475 </s> |
|
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|
2476 <lb/> |
|
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|
2477 <s xml:id="echoid-s174" xml:space="preserve"> |
|
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|
2478 Quoniam: |
|
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|
2479 <lb/>[<emph style="it">tr: |
|
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|
2480 Because: |
|
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|
2481 </emph>]<lb/> |
|
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|
2482 </s> |
|
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|
2483 <lb/> |
|
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|
2484 <s xml:id="echoid-s175" xml:space="preserve"> |
|
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|
2485 Adde utrolique. |
|
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|
2486 <lb/>[<emph style="it">tr: |
|
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|
2487 Add to each side. |
|
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|
2488 </emph>]<lb/> |
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2489 </s> |
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2490 <lb/> |
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2491 <s xml:id="echoid-s176" xml:space="preserve"> |
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|
2492 Ergo. |
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2493 <lb/>[<emph style="it">tr: |
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2494 Therefore. |
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2495 </emph>]<lb/> |
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2496 </s> |
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2497 <lb/> |
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2498 <s xml:id="echoid-s177" xml:space="preserve"> |
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2499 Et ita. |
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2500 <lb/>[<emph style="it">tr: |
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|
2501 And thus. |
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2502 </emph>]<lb/> |
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2503 </s> |
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2504 <lb/> |
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2505 <s xml:id="echoid-s178" xml:space="preserve"> |
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2506 Sit. |
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2507 <lb/>[<emph style="it">tr: |
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2508 Let. |
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2509 </emph>]<lb/> |
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2510 </s> |
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2511 <lb/> |
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2512 <s xml:id="echoid-s179" xml:space="preserve"> |
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|
2513 Dico quod. per Hypobibasmum. |
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2514 <lb/>[<emph style="it">tr: |
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|
2515 I say that, by hypobibasmus. |
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2516 </emph>]<lb/> |
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2517 </s> |
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2518 </p> |
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2519 <p xml:lang=""> |
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2520 <s xml:id="echoid-s180" xml:space="preserve"> |
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2521 Sit. |
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2522 <lb/>[<emph style="it">tr: |
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2523 Let. |
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|
2524 </emph>]<lb/> |
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2525 </s> |
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2526 <lb/> |
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2527 <s xml:id="echoid-s181" xml:space="preserve"> |
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|
2528 Dico quod: per Parabolismum. |
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2529 <lb/>[<emph style="it">tr: |
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|
2530 I say that, by parabolismus. |
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2531 </emph>]<lb/> |
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2532 </s> |
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2533 <lb/> |
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2534 <s xml:id="echoid-s182" xml:space="preserve"> |
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|
2535 Vel, sit: |
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|
2536 <lb/>[<emph style="it">tr: |
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|
2537 Or, let: |
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2538 </emph>]<lb/> |
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2539 </s> |
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2540 <lb/> |
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2541 <s xml:id="echoid-s183" xml:space="preserve"> |
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|
2542 dico quod. |
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2543 <lb/>[<emph style="it">tr: |
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|
2544 I say that. |
|
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|
2545 </emph>]<lb/> |
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2546 </s> |
|
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|
2547 </p> |
|
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|
2548 <pb file="add_6784_f325v" o="325v" n="650"/> |
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|
2549 <pb file="add_6784_f326" o="326" n="651"/> |
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|
2550 <pb file="add_6784_f326v" o="326v" n="652"/> |
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2551 <pb file="add_6784_f327" o="327" n="653"/> |
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2552 <pb file="add_6784_f327v" o="327v" n="654"/> |
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|
2553 <pb file="add_6784_f328" o="328" n="655"/> |
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2554 <pb file="add_6784_f328v" o="328v" n="656"/> |
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|
2555 <pb file="add_6784_f329" o="329" n="657"/> |
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|
2556 <pb file="add_6784_f329v" o="329v" n="658"/> |
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|
2557 <pb file="add_6784_f330" o="330" n="659"/> |
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|
2558 <pb file="add_6784_f330v" o="330v" n="660"/> |
|
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|
2559 <pb file="add_6784_f331" o="331" n="661"/> |
|
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|
2560 <div xml:id="echoid-div45" type="page_commentary" level="2" n="45"> |
|
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|
2561 <p> |
|
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|
2562 <s xml:id="echoid-s184" xml:space="preserve">[<emph style="it">Note: |
|
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|
2563 <p> |
|
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|
2564 <s xml:id="echoid-s184" xml:space="preserve"> |
|
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|
2565 The reference to Apollonius is to pages 5 and 6 of Commandino's edition, |
|
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|
2566 <emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566). |
|
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|
2567 There are also references at the bottom of the page to |
|
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|
2568 Viète an Cardano. |
|
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|
2569 </s> |
|
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|
2570 <lb/> |
|
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|
2571 <s xml:id="echoid-s185" xml:space="preserve"> |
|
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|
2572 The reference to Viète is to <emph style="it">Apollonius Gallus</emph>, Appendix 2, Problem V. |
|
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|
2573 </s> |
|
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|
2574 <lb/> |
|
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|
2575 <quote xml:lang="lat"> |
|
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|
2576 V. Dato triangulo, invenire punctum, a quo ad apices dati trianguli actæ tres lineæ rectæ imperatam teneant rationem. |
|
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|
2577 </quote> |
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2578 <lb/> |
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2579 <quote> |
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2580 Given a triangle, find a point from which there may be drawn three straight lines |
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2581 to the vertices of the given triangle, keeping a fixed ratio. |
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2582 </quote> |
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2583 <lb/> |
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2584 <s xml:id="echoid-s186" xml:space="preserve"> |
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2585 The reference to Cardano is to his <emph style="it">Opus novum de proportionibus</emph>. |
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2586 The relevant Propositions are 154 (though mistakenly described in the <emph style="it">Opus novum</emph> as 144) |
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2587 and 160. |
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2588 </s> |
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2589 <lb/> |
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2590 <quote xml:lang="lat"> |
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2591 Propositio centesimaquadragesimaquarta <lb/> |
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2592 Sint lineæ datæ alia linea adiungatur, ab extremitatibus autem prioris lineæ duæ rectæ in unum punctum concurrant |
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2593 proportionem habentes quam media inter totam & adiectam, ad adiectam erit punctus concursus a puncto |
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2594 extrema lineæ adiectæ distans per lineam mediam. Quod si ab extremo alicuius lineæ æqualis mediæ |
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2595 seu peripheria circuli cuius semidiameter sit media linea duæ lineæ ad prædicta puncta producantur, |
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2596 ipsæ erunt in proportione mediæ ad adiectam. <lb/> |
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2597 Hæc propositio est admirabilis: ... |
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2598 </quote> |
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2599 <lb/> |
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2600 <quote xml:lang="lat"> |
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2601 Propositio centesimasexagesima <lb/> |
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2602 Proposita linea tribusque in ea signis punctum invenire, ex quo ductæ tres lineæ sint in proportionibus datis. |
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2603 </quote> |
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2604 </p> |
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2605 </emph>] |
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2606 <lb/><lb/></s></p></div> |
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2607 <head xml:id="echoid-head86" xml:space="preserve" xml:lang="lat"> |
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2608 5. Appolonius. pag. 5. 6. |
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2609 <lb/>[<emph style="it">tr: |
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2610 Apollonius, pages 5, 6. |
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2611 </emph>]<lb/> |
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2612 </head> |
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2613 <p xml:lang="lat"> |
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2614 <s xml:id="echoid-s188" xml:space="preserve"> |
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2615 Quæsitum: <lb/> |
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2616 ubicunque signatur in periferia punctum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> <lb/> |
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2617 erit; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>b</mi></mstyle></math> : <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>: vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>b</mi></mstyle></math>. |
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2618 <lb/>[<emph style="it">tr: |
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2619 Sought: <lb/> |
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2620 Wherever a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is placed on the circumference, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi><mo>:</mo><mi>h</mi><mi>b</mi><mo>=</mo><mi>c</mi><mo>:</mo><mi>d</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi><mo>:</mo><mi>k</mi><mi>b</mi></mstyle></math>. |
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2621 </emph>]<lb/> |
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2622 </s> |
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2623 </p> |
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2624 <p xml:lang="lat"> |
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2625 <s xml:id="echoid-s189" xml:space="preserve"> |
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2626 sint data puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <lb/> |
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2627 Data ratio. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. <lb/> |
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2628 producatur, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, versus, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> |
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2629 <lb/>[<emph style="it">tr: |
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2630 Let the given points be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, the given ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>:</mo><mi>d</mi></mstyle></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> be produced towards <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. |
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2631 </emph>]<lb/> |
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2632 </s> |
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2633 </p> |
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2634 <p xml:lang="lat"> |
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2635 <s xml:id="echoid-s190" xml:space="preserve"> |
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2636 Dico quod: |
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2637 <lb/>[<emph style="it">tr: |
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2638 I say that: |
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2639 </emph>]<lb/> |
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2640 </s> |
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2641 </p> |
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2642 <p xml:lang="lat"> |
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2643 <s xml:id="echoid-s191" xml:space="preserve"> |
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2644 Inde: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> maior, quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math><lb/> |
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2645 minor, quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> <lb/> |
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2646 fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>k</mi><mo>=</mo><mi>g</mi></mstyle></math> <lb/> |
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2647 fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>h</mi></mstyle></math> periferia <lb/> |
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2648 sumatur quovis puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> <lb/> |
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2649 Ducantur: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>f</mi></mstyle></math>. |
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2650 <lb/>[<emph style="it">tr: |
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2651 Whence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math>, less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>k</mi><mo>=</mo><mi>g</mi></mstyle></math>, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>h</mi></mstyle></math> be the circumference, |
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2652 taking any point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. Let there be drawn <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>f</mi></mstyle></math>. |
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2653 </emph>]<lb/> |
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2654 </s> |
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2655 </p> |
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2656 <p xml:lang="lat"> |
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2657 <s xml:id="echoid-s192" xml:space="preserve"> |
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2658 * Ducantur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>l</mi></mstyle></math>, parallela, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>. <lb/> |
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2659 ubicunque signatur in periferia punctum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> <lb/> |
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2660 erit; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>b</mi></mstyle></math> : <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>: vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>b</mi></mstyle></math>. |
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2661 <lb/>[<emph style="it">tr: |
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2662 Taking <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>l</mi></mstyle></math> parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>, wherever the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is placed on the circumference, |
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2663 then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi><mo>:</mo><mi>h</mi><mi>b</mi><mo>=</mo><mi>c</mi><mo>:</mo><mi>d</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi><mo>:</mo><mi>k</mi><mi>b</mi></mstyle></math>. |
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2664 </emph>]<lb/> |
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2665 </s> |
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2666 </p> |
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2667 <p xml:lang="lat"> |
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2668 <s xml:id="echoid-s193" xml:space="preserve"> |
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2669 Corollaria. <lb/> |
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2670 Hinc a tribus punctis sive sint in recta <lb/> |
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2671 vel non; possunt duci tres lineæ ad unum <lb/> |
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2672 punctum, <emph style="st">ut s</emph> et erunt in data ratione. |
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2673 <lb/>[<emph style="it">tr: |
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2674 Corollary <lb/> |
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2675 Hence from three points, whether in a straight line or not, it is possible to draw three lines to a single point, |
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2676 and they will be in the given ratio. |
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2677 </emph>]<lb/> |
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2678 </s> |
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2679 </p> |
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2680 <p xml:lang="lat"> |
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2681 <s xml:id="echoid-s194" xml:space="preserve"> |
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2682 vide vertam <lb/> |
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2683 in Apolonio gallo <lb/> |
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2684 et card: de prop. pag. 145. 162. |
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2685 <lb/>[<emph style="it">tr: |
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2686 see over, in <emph style="it">Apollonius Gallus</emph>, |
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2687 and Cardano, <emph style="it">De proportionibus</emph>, pages 145, 162. |
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2688 </emph>]<lb/> |
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2689 </s> |
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2690 </p> |
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2691 <pb file="add_6784_f331v" o="331v" n="662"/> |
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2692 <pb file="add_6784_f332" o="332" n="663"/> |
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2693 <div xml:id="echoid-div46" type="page_commentary" level="2" n="46"> |
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2694 <p> |
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2695 <s xml:id="echoid-s195" xml:space="preserve">[<emph style="it">Note: |
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2696 <p> |
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2697 <s xml:id="echoid-s195" xml:space="preserve"> |
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2698 The reference is to pages 5 and 6 of Commandino's edition of Apollonius, |
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2699 <emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566). |
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2700 </s> |
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2701 </p> |
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2702 </emph>] |
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2703 <lb/><lb/></s></p></div> |
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2704 <head xml:id="echoid-head87" xml:space="preserve" xml:lang="lat"> |
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2705 Ad appolonium. pa. 5. 6. |
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2706 <lb/>[<emph style="it">tr: |
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2707 On Apollonius, pages 5, 6 |
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2708 </emph>]<lb/> |
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2709 </head> |
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2710 <p xml:lang="lat"> |
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2711 <s xml:id="echoid-s197" xml:space="preserve"> |
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2712 Data puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math>, in linea, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> <lb/> |
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2713 Invenire lineam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> <lb/> |
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2714 ita ut sit: <lb/> |
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2715 Sit factum: <lb/> |
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2716 Tum: |
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2717 <lb/>[<emph style="it">tr: |
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2718 Given a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math> in a line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, find the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> so that: <lb/> |
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2719 Let it be done, then: |
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2720 </emph>]<lb/> |
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2721 </s> |
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2722 </p> |
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2723 <p xml:lang="lat"> |
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2724 <s xml:id="echoid-s198" xml:space="preserve"> |
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2725 Aliter <lb/> |
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2726 <lb/>[...]<lb/> <lb/> |
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2727 sed idem ut supra |
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2728 <lb/>[<emph style="it">tr: |
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|
2729 Another way <lb/> |
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2730 <lb/>[...]<lb/> <lb/> |
|
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2731 but the same as above |
|
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2732 </emph>]<lb/> |
|
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2733 </s> |
|
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2734 </p> |
|
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2735 <p xml:lang="lat"> |
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2736 <s xml:id="echoid-s199" xml:space="preserve"> |
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2737 Invenire <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>k</mi></mstyle></math> |
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2738 <lb/>[<emph style="it">tr: |
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2739 To find <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>k</mi></mstyle></math>. |
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2740 </emph>]<lb/> |
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2741 </s> |
|
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2742 </p> |
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2743 <pb file="add_6784_f332v" o="332v" n="664"/> |
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2744 <pb file="add_6784_f333" o="333" n="665"/> |
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2745 <pb file="add_6784_f333v" o="333v" n="666"/> |
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2746 <pb file="add_6784_f334" o="334" n="667"/> |
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2747 <pb file="add_6784_f334v" o="334v" n="668"/> |
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2748 <pb file="add_6784_f335" o="335" n="669"/> |
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2749 <pb file="add_6784_f335v" o="335v" n="670"/> |
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2750 <pb file="add_6784_f336" o="336" n="671"/> |
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2751 <pb file="add_6784_f336v" o="336v" n="672"/> |
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2752 <pb file="add_6784_f337" o="337" n="673"/> |
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2753 <pb file="add_6784_f337v" o="337v" n="674"/> |
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2754 <pb file="add_6784_f338" o="338" n="675"/> |
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2755 <pb file="add_6784_f338v" o="338v" n="676"/> |
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2756 <pb file="add_6784_f339" o="339" n="677"/> |
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2757 <pb file="add_6784_f339v" o="339v" n="678"/> |
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2758 <pb file="add_6784_f340" o="340" n="679"/> |
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2759 <pb file="add_6784_f340v" o="340v" n="680"/> |
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2760 <pb file="add_6784_f341" o="341" n="681"/> |
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2761 <pb file="add_6784_f341v" o="341v" n="682"/> |
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2762 <pb file="add_6784_f342" o="342" n="683"/> |
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2763 <pb file="add_6784_f342v" o="342v" n="684"/> |
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2764 <pb file="add_6784_f343" o="343" n="685"/> |
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2765 <pb file="add_6784_f343v" o="343v" n="686"/> |
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2766 <pb file="add_6784_f344" o="344" n="687"/> |
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2767 <pb file="add_6784_f344v" o="344v" n="688"/> |
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2768 <pb file="add_6784_f345" o="345" n="689"/> |
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2769 <pb file="add_6784_f345v" o="345v" n="690"/> |
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2770 <pb file="add_6784_f346" o="346" n="691"/> |
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2771 <pb file="add_6784_f346v" o="346v" n="692"/> |
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2772 <pb file="add_6784_f347" o="347" n="693"/> |
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2773 <pb file="add_6784_f347v" o="347v" n="694"/> |
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2774 <pb file="add_6784_f348" o="348" n="695"/> |
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|
2775 <div xml:id="echoid-div47" type="page_commentary" level="2" n="47"> |
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|
2776 <p> |
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2777 <s xml:id="echoid-s200" xml:space="preserve">[<emph style="it">Note: |
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|
2778 <p> |
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2779 <s xml:id="echoid-s200" xml:space="preserve"> |
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|
2780 On this page Harriot investigates Proposition 18 from Viète's |
|
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2781 <emph style="it">Supplementum geometriæ</emph> (1593). |
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2782 </s> |
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|
2783 <lb/> |
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|
2784 <quote xml:lang="lat"> |
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|
2785 Proposition XVIII. <lb/> |
|
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|
2786 Si duo triangula fuerint aequicrura singula, & ipsa alterum alteri cruribus aequalia, |
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2787 angulus autem qui est ad basin secundi sit triplus anguli qui est ad basin primi: |
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2788 triplum solidum sub quadrato cruris communis & dimidia base primi multata continuatave longitudine |
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2789 ejus cujus quadratum æquale est triplo quadrato altitudinis primi, cum multabitur ejusdem dimidiæ |
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2790 basis multatæ continuatve cubo, æquale est solido sub base secundi & ejusdem cruris quadrato. |
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|
2791 </quote> |
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2792 <lb/> |
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|
2793 <quote> |
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2794 If two triangles are each isosceles, equal to one another in theri legs, |
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2795 and moreover the angle at the base of the second is three times the angle at the base of the first, |
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2796 then three times the product of the square of the common leg and half the base of the first |
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2797 decreased or increased by a length whose square is equal to three times the square of the altitude of the first, |
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2798 when reduced by the cube of the same half base thus decreased or increased, |
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2799 is equal to the product of the second base and the square of the common leg. |
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|
2800 </quote> |
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2801 <lb/> |
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2802 <s xml:id="echoid-s201" xml:space="preserve"> |
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|
2803 For Harriot's statement of Propostion 18, and a geometric version of the proof, see Add MS 6784, f. 349. |
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2804 Here he works the proposition algebraically. |
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2805 </s> |
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2806 <lb/> |
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2807 <s xml:id="echoid-s202" xml:space="preserve"> |
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2808 This page also refers to Proposition 17 from the <emph style="it">Supplementum</emph>, |
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2809 (see MS 6784, f. 350). |
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2810 </s> |
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|
2811 </p> |
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2812 </emph>] |
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|
2813 <lb/><lb/></s></p></div> |
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2814 <head xml:id="echoid-head88" xml:space="preserve"> |
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2815 prop. 18. Supplementi. |
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2816 <lb/>[<emph style="it">tr: |
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2817 Proposition 18 from the Supplementum |
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2818 </emph>]<lb/> |
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2819 </head> |
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2820 <pb file="add_6784_f348v" o="348v" n="696"/> |
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2821 <pb file="add_6784_f349" o="349" n="697"/> |
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2822 <div xml:id="echoid-div48" type="page_commentary" level="2" n="48"> |
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2823 <p> |
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2824 <s xml:id="echoid-s204" xml:space="preserve">[<emph style="it">Note: |
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2825 <p> |
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2826 <s xml:id="echoid-s204" xml:space="preserve"> |
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2827 On this page Harriot investigates Proposition 18 from Viète's |
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2828 <emph style="it">Supplementum geometriæ</emph> (1593). |
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2829 </s> |
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2830 <lb/> |
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2831 <quote xml:lang="lat"> |
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2832 Proposition XVIII. <lb/> |
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2833 Si duo triangula fuerint aequicrura singula, & ipsa alterum alteri cruribus aequalia, |
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2834 angulus autem qui est ad basin secundi sit triplus anguli qui est ad basin primi: |
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2835 triplum solidum sub quadrato cruris communis & dimidia base primi multata continuatave longitudine |
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2836 ejus cujus quadratum æquale est triplo quadrato altitudinis primi, cum multabitur ejusdem dimidiæ |
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2837 basis multatæ continuatve cubo, æquale est solido sub base secundi & ejusdem cruris quadrato. |
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2838 </quote> |
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2839 <lb/> |
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2840 <quote> |
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2841 If two triangles are each isosceles, equal to one another in theri legs, |
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2842 and moreover the angle at the base of the second is three times the angle at the base of the first, |
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2843 then three times the product of the square of the common leg and half the base of the first |
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2844 decreased or increased by a length whose square is equal to three times the square of the altitude of the first, |
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2845 when reduced by the cube of the same half base thus decreased or increased, |
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2846 is equal to the product of the second base and the square of the common leg. |
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2847 </quote> |
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2848 <lb/> |
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2849 <s xml:id="echoid-s205" xml:space="preserve"> |
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2850 This page refers to several previous propositions from the <emph style="it">Supplementum</emph>, |
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2851 namely Proposition 12 and 14b (Add MS 6784, f. 353), |
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2852 Proposition 16 (add MS 6784, f. 351) and Proposition 17 (add MS 6784, f. 350). |
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2853 </s> |
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2854 </p> |
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2855 </emph>] |
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2856 <lb/><lb/></s></p></div> |
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2857 <head xml:id="echoid-head89" xml:space="preserve"> |
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2858 prop. 18. Supplementi. |
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2859 <lb/>[<emph style="it">tr: |
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2860 Proposition 18 from the Supplementum |
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2861 </emph>]<lb/> |
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2862 </head> |
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2863 <p xml:lang="lat"> |
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2864 <s xml:id="echoid-s207" xml:space="preserve"> |
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2865 Si duo triangula fuerint aequicrura singula, et ipsa alterum alteri cruribus aequalia; angulus <lb/> |
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2866 autem qui est ad basin secundi sit triplus anguli qui est ad basin primi. Triplum solidum <lb/> |
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2867 sub quadrato cruris communis, et dimidia base primi multata continuatave longitudine <lb/> |
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2868 ejus cujus quadratum æquale est triplo quadrato altitudinis primi, cum multabitur ejusdem <lb/> |
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2869 dimidiæ basis multatæ continuatve cubo, æquale est solido sub base secundi et ejusdem <lb/> |
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2870 cruris quadrato. |
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2871 <lb/>[<emph style="it">tr: |
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2872 If two triangles are each isosceles, equal to one another in their legs, |
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2873 and moreover the angle at the base of the second is three times the angle at the base of the first, |
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2874 then three times the product of the square of the common leg and half the base of the first |
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2875 decreased or increased by a length whose square is equal to three times the square of the altitude of the first, |
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2876 when reduced by the cube of the same half base thus decreased or increased, |
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2877 is equal to the product of the second base and the square of the common leg. |
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2878 </emph>]<lb/> |
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2879 </s> |
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2880 </p> |
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2881 <p xml:lang="lat"> |
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2882 <s xml:id="echoid-s208" xml:space="preserve"> |
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2883 Sit triangulum primum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math>, secundum <lb/> |
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2884 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>D</mi><mi>E</mi></mstyle></math>. quorum crura et anguli sint <lb/> |
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2885 ut exigit propositio. Et sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>B</mi></mstyle></math> dupla <lb/> |
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2886 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>F</mi></mstyle></math>. Tum quadratum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>F</mi></mstyle></math> erit triplum quadrati <lb/> |
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2887 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>F</mi></mstyle></math>  <lb/> |
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2888 Dico |
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2889 <lb/>[<emph style="it">tr: |
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2890 Let the first triangle be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math> and the second <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>D</mi><mi>E</mi></mstyle></math>, whose sides and angles are as specified in the proposition. |
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2891 And let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>B</mi></mstyle></math> be twice <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>F</mi></mstyle></math>. Then the square of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>F</mi></mstyle></math> is three times the square of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>F</mi></mstyle></math>. |
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2892 </emph>]<lb/> |
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2893 </s> |
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2894 </p> |
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2895 <p xml:lang="lat"> |
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2896 <s xml:id="echoid-s209" xml:space="preserve"> |
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2897 Nam: <lb/> |
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2898 per 15,p <lb/>[...]<lb/> Hoc est, in notis proportionalium quas notum 12,p <lb/> |
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2899 1<emph style="super">o</emph>. Ducantur omnia per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> <lb/> |
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2900 <lb/>[...]<lb/> <lb/> |
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2901 Hoc est in notis 12,p. |
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2902 <lb/>[<emph style="it">tr: |
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2903 For by Proposition 15 |
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2904 <lb/>[...]<lb/> that is, in the notation for proportionals noted in Proposition 12, <lb/> |
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2905 1. Multiply everything by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math>. <lb/> |
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2906 <lb/>[...]<lb/> <lb/> |
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2907 That is, in the notation of Proposition 12 |
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2908 </emph>]<lb/> |
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2909 </s> |
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2910 </p> |
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2911 <p xml:lang="lat"> |
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2912 <s xml:id="echoid-s210" xml:space="preserve"> |
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2913 2<emph style="super">o</emph>. Ducantur omnia per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>G</mi></mstyle></math> <lb/> |
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2914 <lb/>[...]<lb/> <lb/> |
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2915 Hoc est in notis 12,p. |
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2916 <lb/>[<emph style="it">tr: |
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2917 2. Multiply everything by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>G</mi></mstyle></math>. <lb/> |
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2918 <lb/>[...]<lb/> <lb/> |
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2919 That is, in the notation of Proposition 12 |
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2920 </emph>]<lb/> |
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2921 </s> |
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2922 </p> |
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2923 <p xml:lang="lat"> |
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2924 <s xml:id="echoid-s211" xml:space="preserve"> |
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2925 Deinde per 16.p <lb/> |
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2926 Hoc est in notis 12,p. <lb/> |
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2927 Sed: per consect: 14.p <lb/> |
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2928 Ergo patet propositum |
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2929 <lb/>[<emph style="it">tr: |
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2930 Thence by Proposition 16, <lb/> |
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2931 That is, in the notation of Proposition 12 <lb/> |
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2932 But by the consequence of Proposition 14, <lb/> |
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2933 Thus the propostion is shown. |
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2934 </emph>]<lb/> |
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2935 </s> |
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2936 </p> |
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2937 <p xml:lang="lat"> |
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2938 <s xml:id="echoid-s212" xml:space="preserve"> |
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2939 Cum 16<emph style="super">a</emph> et 17<emph style="super">a</emph> prop. basis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> notabatur (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>) |
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2940 ideo eius partes <lb/> |
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2941 Scilicet <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>C</mi></mstyle></math> alijs vocalibus notandæ sunt. pro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> nota (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>) <lb/> |
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2942 et pro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>C</mi></mstyle></math>, (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>). <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>E</mi></mstyle></math> servent easdem notas quas ibi <lb/> |
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2943 habuerunt. Videlicet <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>, (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>) et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>E</mi></mstyle></math>, (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>). <lb/> |
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2944 Propositum igitur simplicibus notis ita significatur: |
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2945 <lb/>[<emph style="it">tr: |
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2946 Since in Propositions 16 adn 17, the base <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, therefore its parts, |
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2947 namely <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>C</mi></mstyle></math> may be denoted by other names; |
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2948 for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> put the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> and for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>C</mi></mstyle></math> the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. |
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2949 For <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>E</mi></mstyle></math> use the same notation as they had there, namely <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mo>=</mo><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>E</mi><mo>=</mo><mi>c</mi></mstyle></math>. <lb/> |
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2950 In simple notation the proposition may therefore be written: |
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2951 </emph>]<lb/> |
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2952 </s> |
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2953 </p> |
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2954 <p xml:lang="lat"> |
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2955 <s xml:id="echoid-s213" xml:space="preserve"> |
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2956 igitur: <lb/> |
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2957 Quando æquatio est sub ista <lb/> |
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2958 forma: <lb/> |
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2959 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> erit duplex vel. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math>. vel. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>C</mi></mstyle></math>. |
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2960 <lb/>[<emph style="it">tr: |
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2961 When the equation is in this form, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is twofold, either <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>C</mi></mstyle></math>. |
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2962 </emph>]<lb/> |
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2963 </s> |
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2964 </p> |
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2965 <pb file="add_6784_f349v" o="349v" n="698"/> |
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2966 <pb file="add_6784_f350" o="350" n="699"/> |
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2967 <div xml:id="echoid-div49" type="page_commentary" level="2" n="49"> |
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2968 <p> |
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2969 <s xml:id="echoid-s214" xml:space="preserve">[<emph style="it">Note: |
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2970 <p> |
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2971 <s xml:id="echoid-s214" xml:space="preserve"> |
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2972 On this page Harriot investigates Proposition 17 from Viète's |
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2973 <emph style="it">Supplementum geometriæ</emph> (1593). |
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2974 </s> |
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2975 <lb/> |
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2976 <quote xml:lang="lat"> |
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2977 Proposition XVII. <lb/> |
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2978 Si duo triangula fuerint aequicrura singula, & ipsa alterumalteria cruribus aequalia, |
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2979 angulus autem, quem is qui est ad basin secundi relinquit e duobus rectis, |
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2980 sit triplus anguli qui est ad basin primi: solidum triplum sub base primi & cruris communis quadrato, |
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2981 minus cubo e base primi, aequale est solido sub base secundi & cruris communis quadrato. |
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2982 </quote> |
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2983 <lb/> |
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2984 <quote> |
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2985 If two triangles are each isosceles, both with equal legs, |
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2986 and moreover the angle at the base of the second subtracted from two right angles is |
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2987 three times the angle at the base of the first, |
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2988 then three times the product of the base of the first and the square of the common side, |
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2989 minus the cube of the first base, is equal to the product of the second base and the square of the common side. |
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2990 </quote> |
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2991 <lb/> |
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2992 <s xml:id="echoid-s215" xml:space="preserve"> |
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2993 The working contains reference to three propositions from Euclid's <emph style="it">Elements</emph>. |
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2994 </s> |
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2995 <lb/> |
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2996 <quote> |
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2997 II.6 If a straight line be bisected and produced to any point, |
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2998 the rectangle contained by the whole line so increased, and the part produced, |
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2999 together with the square of half the line, is equal to the square of the line made up of the half, |
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3000 and the produced part. |
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3001 </quote> |
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3002 <lb/> |
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3003 <quote> |
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3004 III.36 If from a point without a circle two straight lines be drawn to it, |
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3005 one of which is a tangent to the circle, and the other cuts it; |
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3006 the rectangle under the whole cutting line and the external segment is equal to the square of the tangent. |
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3007 </quote> |
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3008 <lb/> |
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3009 <quote> |
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3010 I. 47 In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the sides. |
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3011 </quote> |
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3012 </p> |
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3013 </emph>] |
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3014 <lb/><lb/></s></p></div> |
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3015 <head xml:id="echoid-head90" xml:space="preserve"> |
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3016 prop. 17. Supplementi. |
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3017 <lb/>[<emph style="it">tr: |
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3018 Proposition 17 from the Supplementum |
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3019 </emph>]<lb/> |
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3020 </head> |
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3021 <p xml:lang="lat"> |
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3022 <s xml:id="echoid-s217" xml:space="preserve"> |
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3023 Si duo triangula fuerint aequicrura singula, <lb/> |
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3024 et ipsa alterumalteria cruribus aequalia; angulus <lb/> |
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3025 autem, quem is qui est ad basin secundi relinquit <lb/> |
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3026 e duobus rectis, sit triplus anguli qui est ad basin <lb/> |
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3027 <emph style="st">secundi</emph> <emph style="super">primi</emph>. Solidum triplum sub base primi et cruris <lb/> |
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3028 communis quadrato, minus cubo e base primi: aequale <lb/> |
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3029 est solido sub base secundiet cruris communis <lb/> |
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3030 quadrato. |
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3031 <lb/>[<emph style="it">tr: |
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3032 If two triangles are each isosceles, the legs of one equal to the legs of the other, |
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3033 and moreover the angle at the base of the second is three times the angle at the base of the first, |
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3034 then the cube of the first base, minus three times the product of the base of the first and the square of the common side, |
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3035 is equal to the product of the second base and the square of the same side. |
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3036 </emph>]<lb/> |
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3037 </s> |
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3038 </p> |
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3039 <p xml:lang="lat"> |
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3040 <s xml:id="echoid-s218" xml:space="preserve"> |
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3041 per 6,2 el. <lb/> |
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3042 per 36,3 el. <lb/> |
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3043 per 47,1 el. <lb/> |
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3044 <lb/>[...]<lb/> <lb/> |
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3045 quia parallogramma æquialta <lb/> |
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3046 et sunt ut bases. <lb/> |
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3047 <lb/>[...]<lb/> <lb/> |
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3048 vel per notas <lb/> |
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3049 simplices <lb/> |
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3050 Hæque Resoluatur Analogia, erit: <lb/> |
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3051 Propositum |
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3052 <lb/>[<emph style="it">tr: |
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3053 by Elements II.6 <lb/> |
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3054 by Elements III.35 <lb/> |
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3055 by Elements I.47 <lb/> |
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|
3056 <lb/>[...]<lb/> <lb/> |
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3057 because the parallelograms are of equal height and are as the bases. <lb/> |
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3058 <lb/>[...]<lb/> <lb/> |
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3059 or in simple notation <lb/> |
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3060 And this ratio is resolved, hence the proposition: |
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3061 </emph>]<lb/> |
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3062 </s> |
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3063 </p> |
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3064 <pb file="add_6784_f350v" o="350v" n="700"/> |
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3065 <pb file="add_6784_f351" o="351" n="701"/> |
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3066 <div xml:id="echoid-div50" type="page_commentary" level="2" n="50"> |
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3067 <p> |
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3068 <s xml:id="echoid-s219" xml:space="preserve">[<emph style="it">Note: |
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3069 <p> |
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3070 <s xml:id="echoid-s219" xml:space="preserve"> |
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3071 On this page Harriot investigates Proposition 16 from Viète's |
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3072 <emph style="it">Supplementum geometriæ</emph> (1593). |
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3073 </s> |
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3074 <lb/> |
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3075 <quote xml:lang="lat"> |
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3076 Proposition XVI. <lb/> |
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3077 Si duo triangula fuerint aequicrura singula, & ipsa alterum alteri cruribus aequalia, |
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3078 angulus autem qui est ad basin secundi sit triplus anguli qui est ad basin primi: |
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3079 cubus ex base primi, minus triplo solido sub base primi & cruris communis quadrato, |
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3080 aequalis est solido sub base secundi & ejusdem cruris quadrato. |
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3081 </quote> |
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3082 <lb/> |
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3083 <quote> |
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3084 If two triangles are each isosceles, the legs of one equal to the legs of the other, |
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3085 and moreover the angle at the base of the second is three times the angle at the base of the first, |
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3086 then the cube of the first base, minus three times the product of the base of the first and the square of the common side, |
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3087 is equal to the product of the second base and the square of the same side. |
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3088 </quote> |
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3089 <lb/> |
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3090 <s xml:id="echoid-s220" xml:space="preserve"> |
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3091 The working contains a reference to Euclid's <emph style="it">Elements</emph>, Proposition II.5. |
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3092 </s> |
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3093 <lb/> |
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3094 <quote> |
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3095 II.5 If a straight line be divided into two equal parts and also into two unequal parts, |
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3096 the rectangle contained by the unequal parts, |
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3097 together with the square of the line between the points of section, |
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3098 is equal to the square of half that line. |
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3099 </quote> |
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3100 </p> |
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3101 </emph>] |
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3102 <lb/><lb/></s></p></div> |
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3103 <head xml:id="echoid-head91" xml:space="preserve"> |
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3104 prop. 16. Supplementi. |
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3105 <lb/>[<emph style="it">tr: |
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3106 Proposition 16 from the Supplementum |
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3107 </emph>]<lb/> |
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3108 </head> |
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3109 <p xml:lang="lat"> |
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3110 <s xml:id="echoid-s222" xml:space="preserve"> |
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3111 Si duo triangula fuerint aequicrura singula, <lb/> |
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3112 et ipsa alterum alteri cruribus aequalia: angulus <lb/> |
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3113 autem qui est ad basin secundi sit triplus <lb/> |
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3114 anguli qui est ad basin primi. Cubus ex <lb/> |
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3115 base primi, minus triplo solido sub base primi <lb/> |
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3116 et cruris communis quadrato, aequalis <lb/> |
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3117 est solido sub base secundi et ejusdem <lb/> |
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3118 cruris quadrato. |
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3119 <lb/>[<emph style="it">tr: |
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3120 If two triangles are each isosceles, the legs of one equal to the legs of the other, |
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3121 and moreover the angle at the base of the second is three times the angle at the base of the first, |
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3122 then the cube of the first base, minus three times the product of the base of the first and the square of the common side, |
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3123 is equal to the product of the second base and the square of the same side. |
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3124 </emph>]<lb/> |
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3125 </s> |
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3126 </p> |
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3127 <p xml:lang="lat"> |
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3128 <s xml:id="echoid-s223" xml:space="preserve"> |
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3129 per 5,2 el. <lb/> |
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3130 <lb/>[...]<lb/> <lb/> |
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3131 Quia parallogramma æquialta <lb/> |
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3132 et sunt ut bases. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>H</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>D</mi></mstyle></math>. <lb/> |
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3133 <lb/>[...]<lb/> <lb/> |
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3134 vel per notas <lb/> |
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3135 simplices <lb/> |
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3136 Resoluatur analogia et erit: <lb/> |
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3137 Propositum |
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3138 <lb/>[<emph style="it">tr: |
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3139 by Elements II.5 <lb/> |
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3140 <lb/>[...]<lb/> <lb/> |
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|
3141 Because the parallelograms are of equal height and are as the bases <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>H</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>D</mi></mstyle></math>. <lb/> |
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3142 <lb/>[...]<lb/> <lb/> |
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3143 or in simple notation <lb/> |
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3144 The ratio is resolved, and hence the proposition: |
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3145 </emph>]<lb/> |
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3146 </s> |
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3147 </p> |
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3148 <pb file="add_6784_f351v" o="351v" n="702"/> |
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3149 <pb file="add_6784_f352" o="352" n="703"/> |
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3150 <div xml:id="echoid-div51" type="page_commentary" level="2" n="51"> |
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3151 <p> |
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3152 <s xml:id="echoid-s224" xml:space="preserve">[<emph style="it">Note: |
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3153 <p> |
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3154 <s xml:id="echoid-s224" xml:space="preserve"> |
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3155 On this page Harriot investigates Proposition 15 from Viète's |
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3156 <emph style="it">Supplementum geometriæ</emph> (1593). |
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3157 </s> |
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3158 <lb/> |
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3159 <quote xml:lang="lat"> |
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3160 Proposition XV. <lb/> |
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3161 Si e circumferential circuli cadant in diametrum perpendiculares duæ, una in centro, altera extra centrum; |
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3162 & ad perpendicularem in centro agatur ex puncto incidentiæ perpendicularis alterius, |
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3163 linea recta faciens cum diametro angulum æqualem trienti recti; |
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3164 a puncto autem quo acta illa secat perpendiculare in centro, ducatur alia linea recta ad angulum semicirculi: |
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3165 triplum quadratum huius, æquale est tam quadrato perpendicularis quae incidit extra centrum, |
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3166 quam quadratis segmentorum diametri, inter quæ perpendicularis illa media est proportionalis. |
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3167 </quote> |
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3168 <lb/> |
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3169 <quote> |
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3170 If from the circumference of a circle there fall two perpendiculars onto the diameter, |
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3171 one to the centre, the other off-centre; and to the perpendicular to the centre there is drawn |
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3172 from the point of incidence of the other perpendicular a straight line making an angle equal to |
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3173 one-third of a right angle to the diameter; moreover from the point where that line cuts the perpendicular to the centre, |
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3174 there is drawn another line to the angle of the semicircle, then three times the square of it |
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3175 is equal to the square of the perpendicular which falls off-centre |
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3176 and the squares of the segments of the diameter between which the perpendicular is the mean proportional. |
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3177 </quote> |
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3178 <lb/> |
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3179 <s xml:id="echoid-s225" xml:space="preserve"> |
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3180 The working contains a reference to Euclid's <emph style="it">Elements</emph>, Proposition II.4. |
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3181 </s> |
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3182 <lb/> |
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3183 <quote> |
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3184 II.4 If a straight line be divided into any two parts, |
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3185 the square of the whole line is equal to the squares of the parts, |
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3186 together with twice the rectangle contained by the parts. |
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3187 </quote> |
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3188 </p> |
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3189 </emph>] |
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3190 <lb/><lb/></s></p></div> |
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3191 <head xml:id="echoid-head92" xml:space="preserve"> |
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3192 prop. 15. Supplementi |
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3193 <lb/>[<emph style="it">tr: |
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3194 Proposition 15 from the Supplementum |
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3195 </emph>]<lb/> |
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3196 </head> |
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3197 <p xml:lang="lat"> |
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3198 <s xml:id="echoid-s227" xml:space="preserve"> |
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3199 Si e circumferential circuli cadant in <lb/> |
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3200 diametrum perpendiculares duæ; una in <lb/> |
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3201 centro; altera extra centrum: et ad per-<lb/> |
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3202 pendicularem in centro agatur ex puncto <lb/> |
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3203 incidentiæ perpendicularis alterius, linea <lb/> |
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3204 recta faciens cum diametro angulum æqualem <lb/> |
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3205 trienti recti, a puncto autem quo acta illa secat <lb/> |
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3206 perpendiculare in centro, ducatur alia <lb/> |
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3207 linea recta ad angulum semicirculi; Triplum <lb/> |
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3208 quadratum huius, æquale est tam quadrato perpendicularis quae incidit extra centrum, <lb/> |
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3209 quam quadratis segmentorum diametri, inter quæ perpendicularis illa media est <lb/> |
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3210 proportionalis. |
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3211 <lb/>[<emph style="it">tr: |
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3212 If from the circumference of a circle there fall two perpendiculars onto the diameter, |
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3213 one to the centre, the other off-centre; and to the perpendicular to the centre there is drawn |
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3214 from the point of incidence of the other perpendicular a straight line making an angle equal to |
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3215 one-third of a right angle to the diameter; moreover from the point where that line cuts the perpendicular to the centre, |
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3216 there is drawn another line to the angle of the semicircle, then three times the square of it |
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3217 is equal to the square of the perpendicular which falls off-centre |
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3218 and the squares of the segments of the diameter between which the perpendicular is the mean proportional. |
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3219 </emph>]<lb/> |
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3220 </s> |
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3221 <lb/> |
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3222 <s xml:id="echoid-s228" xml:space="preserve"> |
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3223 Sit diameter circuli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math>, a cuius circumferentia cadat perpendiculariter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>B</mi></mstyle></math> et fit <lb/> |
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3224 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> minus segmentum, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> maius, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> verum centro. Sed et cadat quoque e circumferentia <lb/> |
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3225 perpendiculariter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>E</mi></mstyle></math>, et ex <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> ducatur recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>G</mi></mstyle></math> ita ut angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>B</mi><mi>E</mi></mstyle></math> sit æqualis trienti <lb/> |
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3226 recti, unde fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>G</mi></mstyle></math> dupla ipsius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>E</mi></mstyle></math>; et iungatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math>. Dico triplum quadratum ex <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> <lb/> |
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3227 æquari quadrato ex <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>B</mi></mstyle></math>, una cum quadrato ex <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> et quadrato ex <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math>. |
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3228 <lb/>[<emph style="it">tr: |
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3229 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math> be the diameter of a circle, from whose circumference there falls perpendicularly <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>B</mi></mstyle></math>, |
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3230 and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> be the lesser segment, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> the greater, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> the centre. |
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3231 But there also falls perpendicularly from the circumference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>E</mi></mstyle></math>, and from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> there is drawn a line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>G</mi></mstyle></math> |
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3232 so that the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>B</mi><mi>E</mi></mstyle></math> is equal to a third of a right angle, whence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>G</mi></mstyle></math> is twice <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>E</mi></mstyle></math>; and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> is joined. |
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3233 I say that three times the square on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> is equal to the square on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>B</mi></mstyle></math> |
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3234 together with the square on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> and the squareon <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math>. |
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3235 </emph>]<lb/> |
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3236 </s> |
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3237 <lb/> |
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3238 <s xml:id="echoid-s229" xml:space="preserve"> |
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3239 Etiam <lb/> |
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3240 per 4,2 El. <lb/> |
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3241 <lb/>[...]<lb/> |
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3242 Addatur utrovisque <lb/> |
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3243 <lb/>[...]<lb/> |
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3244 Ergo <lb/> |
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3245 propositum |
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3246 <lb/>[<emph style="it">tr: |
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3247 Also by Elements II.4 <lb/> |
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3248 <lb/>[...]<lb/> <lb/> |
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3249 Hence the proposition |
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3250 </emph>]<lb/> |
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3251 </s> |
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3252 </p> |
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3253 <head xml:id="echoid-head93" xml:space="preserve"> |
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3254 Hinc tale Consectarium potest efferri |
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3255 <lb/>[<emph style="it">tr: |
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3256 Here a Consequence of this kind may be inferred |
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3257 </emph>]<lb/> |
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3258 </head> |
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3259 <p xml:lang="lat"> |
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3260 <s xml:id="echoid-s230" xml:space="preserve"> |
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3261 Datis tribus continue proportionalibus: invenire lineam cuius <lb/> |
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3262 quadratum sit tertia pars adgregati quadratorum e tribus <lb/> |
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3263 proportionalibus. |
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3264 <lb/>[<emph style="it">tr: |
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3265 Given three continued proportionals, |
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3266 find a line whose square is a third of the sum of the squares of all three proportionals. |
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3267 </emph>]<lb/> |
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3268 </s> |
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3269 </p> |
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3270 <pb file="add_6784_f352v" o="352v" n="704"/> |
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3271 <pb file="add_6784_f353" o="353" n="705"/> |
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3272 <div xml:id="echoid-div52" type="page_commentary" level="2" n="52"> |
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3273 <p> |
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3274 <s xml:id="echoid-s231" xml:space="preserve">[<emph style="it">Note: |
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3275 <p> |
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3276 <s xml:id="echoid-s231" xml:space="preserve"> |
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3277 On this page Harriot investigates Propositions 12, 13, and 14 from Viète's |
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3278 <emph style="it">Supplementum geometriæ</emph> (1593). |
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3279 </s> |
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3280 <lb/> |
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3281 <quote xml:lang="lat"> |
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3282 Proposition XII. <lb/> |
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3283 Si fuerint tres lineæ rectæ proportionales: cubus compositæ e duabus extremis, |
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3284 minus solido quod fit sub eadem composita & adgregato quadratorum a tribus, |
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3285 æqualis est solido sub eadem composita & quadrato secundæ. |
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3286 </quote> |
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3287 <lb/> |
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3288 <quote> |
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3289 If there are three proportional lines, the cube of the sum of the two extremes, |
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3290 minus the product of that sum and the sum of squares of all three, |
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3291 is equal to the product of the sum and the square of the second. |
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3292 </quote> |
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3293 <lb/> |
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3294 <quote xml:lang="lat"> |
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3295 Proposition XIII. <lb/> |
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3296 Si fuerint tres lineæ rectæ proportionales: solidum sub prima & adgregato quadratorum a tribus, |
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3297 minus cubo e prima, æquale est solido sub eadem prima & adgregato quadratorum secundæ & tertiæ. |
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3298 </quote> |
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3299 <lb/> |
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3300 <quote> |
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3301 If there are three proportional lines, the product of the first and the sum of squares of all three, |
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|
3302 minus the cube of the first, is equal to the product of the first and the sum of squares of the second and third. |
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3303 </quote> |
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3304 <lb/> |
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3305 <quote xml:lang="lat"> |
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3306 Proposition XIV. <lb/> |
|
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3307 Si fuerint tres lineæ rectæ proportionales: solidum sub prima & adgregatum quadratorum a tribus, |
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3308 minus cubo e tertia, æquale est solido sub eadem tertia & adgregato quadratorum primæ & secundæ. |
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3309 </quote> |
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3310 <lb/> |
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3311 <quote> |
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3312 If there are three proportional lines, the product of the first and the sum of squares of all three, |
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3313 minus the cube of the third, is equal to the product of the third and the sum of the first and second. |
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3314 </quote> |
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3315 <lb/> |
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3316 <s xml:id="echoid-s232" xml:space="preserve"> |
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3317 The 'Consectarium' appears verbally in Viete's proposition; Harriot has re-written it in symbolic notation. |
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3318 </s> |
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3319 </p> |
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3320 </emph>] |
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3321 <lb/><lb/></s></p></div> |
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3322 <head xml:id="echoid-head94" xml:space="preserve"> |
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3323 prop. 12. Supplementi |
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3324 <lb/>[<emph style="it">tr: |
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3325 Proposition 12 from the Supplementum |
|
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3326 </emph>]<lb/> |
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3327 </head> |
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3328 <p xml:lang="lat"> |
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3329 <s xml:id="echoid-s234" xml:space="preserve"> |
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3330 Si fuerint tres lineæ rectæ proportionales: cubus compositæ e duabus extremis, <lb/> |
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3331 minus solido quod fit sub eadem composita et adgregato quadratorum a tribus: <lb/> |
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3332 æqualis est solido sub eadem composita et quadrato secundæ. |
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3333 <lb/>[<emph style="it">tr: |
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3334 If there are three proportional lines, the cube of the sum of the two extremes, |
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3335 minus the product of that sum and the sum of squares of all three, |
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3336 is equal to the product of the sum and the square of the second. |
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3337 </emph>]<lb/> |
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3338 </s> |
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3339 <lb/> |
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3340 <s xml:id="echoid-s235" xml:space="preserve"> |
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3341 Sint 3 continue proportionales <lb/> |
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3342 utrinque addatur <lb/> |
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3343 <lb/>[...]<lb/> <lb/> |
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3344 Fiant solida ab extremis et etiam a medijs, et inde: <lb/> |
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3345 propositum |
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3346 <lb/>[<emph style="it">tr: |
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3347 let there be three continued proportionals <lb/> |
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3348 add to each side <lb/> |
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3349 <lb/>[...]<lb/> <lb/> |
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3350 There may be made solids from the extremes and also form the means, and hence the proposition: |
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3351 </emph>]<lb/> |
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3352 </s> |
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3353 </p> |
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3354 <p xml:lang="lat"> |
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3355 <s xml:id="echoid-s236" xml:space="preserve"> |
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3356 Prop. 13. Si fuerint tres lineæ rectæ proportionales: solidum sub prima et adgregato <lb/> |
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3357 quadratorum tribus, minus cubo e prima: æquale est solido sub eadem <lb/> |
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3358 prima et adgregato quadratorum secundæ et tertiæ. |
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3359 <lb/>[<emph style="it">tr: |
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|
3360 Proposition 13. If there are three proportional lines, the product of the first and the sum of squares of all three, |
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3361 minus the cube of the first, is equal to the product of the first and the sum of squares of the second and third. |
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3362 </emph>]<lb/> |
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3363 </s> |
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3364 <lb/> |
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3365 <s xml:id="echoid-s237" xml:space="preserve"> |
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3366 Sint tres continue proportionales <lb/> |
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3367 <lb/>[...]<lb/> <lb/> |
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3368 Resoluatur Analogia et erit: <lb/> |
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3369 Propositum |
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3370 <lb/>[<emph style="it">tr: |
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|
3371 Let there be three continued proportionals <lb/> |
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3372 <lb/>[...]<lb/> <lb/> |
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3373 The ratio is resolved, and hence the proposition: |
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3374 </emph>]<lb/> |
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|
3375 </s> |
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3376 </p> |
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3377 <p xml:lang="lat"> |
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3378 <s xml:id="echoid-s238" xml:space="preserve"> |
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|
3379 Prop. 14. Si fuerint tres lineæ rectæ proportionales: solidum sub prima et adgregatum quadratorum <lb/> |
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3380 a tribus minus cubo e tertia: æquale est solido sub eadem tertia et adgregato <lb/> |
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|
3381 quadratorum primæ et secundæ. |
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3382 <lb/>[<emph style="it">tr: |
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|
3383 Proposition 14. If there are three proportional lines, the product of the first and the sum of squares of all three, |
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|
3384 minus the cube of the third, is equal to the product of the third and the sum of the first and second. |
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|
3385 </emph>]<lb/> |
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3386 </s> |
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|
3387 <lb/> |
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3388 <s xml:id="echoid-s239" xml:space="preserve"> |
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|
3389 Sint tres continue proportionales <lb/> |
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3390 <lb/>[...]<lb/> <lb/> |
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3391 Resoluatur Analogia et erit: <lb/> |
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3392 Propositum |
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3393 <lb/>[<emph style="it">tr: |
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3394 Let there be three continued proportionals <lb/> |
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3395 <lb/>[...]<lb/> <lb/> |
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3396 The ratio is resolved, and hence the proposition: |
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3397 </emph>]<lb/> |
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3398 </s> |
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3399 </p> |
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3400 <head xml:id="echoid-head95" xml:space="preserve"> |
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3401 Consectarium |
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3402 <lb/>[<emph style="it">tr: |
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3403 Consequence |
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3404 </emph>]<lb/> |
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3405 </head> |
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3406 <p xml:lang="lat"> |
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3407 <s xml:id="echoid-s240" xml:space="preserve"> |
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3408 Quia æquantur æqualibus <lb/> |
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3409 ex antecedente consectario. |
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3410 <lb/>[<emph style="it">tr: |
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3411 Because equals are equated to equals, by the preceding conclusion. |
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3412 </emph>]<lb/> |
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3413 </s> |
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3414 </p> |
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3415 <pb file="add_6784_f353v" o="353v" n="706"/> |
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3416 <pb file="add_6784_f354" o="354" n="707"/> |
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3417 <div xml:id="echoid-div53" type="page_commentary" level="2" n="53"> |
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3418 <p> |
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3419 <s xml:id="echoid-s241" xml:space="preserve">[<emph style="it">Note: |
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3420 <p> |
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3421 <s xml:id="echoid-s241" xml:space="preserve"> |
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3422 On this page Harriot investigates Propositions 10 and 11 from Viète's |
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3423 <emph style="it">Supplementum geometriæ</emph> (1593). |
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3424 </s> |
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3425 <lb/> |
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3426 <quote xml:lang="lat"> |
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3427 Proposition X. <lb/> |
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3428 Si fuerint tres lineæ rectæ proportionales: est ut prima ad tertiam, |
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3429 ita adgregatum quadratorum primæ & secundæ ad adgregatum quadratorum secundæ & tertiæ. |
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3430 </quote> |
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3431 <lb/> |
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3432 <quote> |
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3433 If there are three proportional lines, as the first is to the third, |
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3434 so is the sum of squares of the first and second to the sum of squares of the second and third. |
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3435 </quote> |
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3436 <lb/> |
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3437 <quote xml:lang="lat"> |
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3438 Proposition XI. <lb/> |
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3439 Si fuerint tres lineæ rectæ proportionales: est ut prima ad adgregatum primae & tertiæ, |
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3440 ita quadratum secundæ ad adgregatum quadratorum secundæ & tertiæ. |
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3441 </quote> |
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3442 <lb/> |
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3443 <quote> |
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3444 If there are three proportional lines, as the first is to the sum of the first and third, |
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3445 so is the square of the second to the sum of squares of the second and third. |
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3446 </quote> |
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3447 <lb/> |
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3448 <s xml:id="echoid-s242" xml:space="preserve"> |
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3449 There are two references to Euclid's <emph style="it">Elements</emph>, Proposition VI.20. |
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3450 </s> |
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3451 <lb/> |
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3452 <quote> |
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3453 VI.20 Similar polygons my be divided into the same number of similar triangles, |
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3454 each similar pair of which are proportional to the polygons; |
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3455 and the polygons are to each other in the duplicate ratio of their homologous sides. |
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3456 </quote> |
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3457 <lb/> |
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3458 <s xml:id="echoid-s243" xml:space="preserve"> |
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3459 The 'Consectarium' appears verbally in Viete's proposition; Harriot has reinterpreted it symbolically. |
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3460 </s> |
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3461 </p> |
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3462 </emph>] |
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3463 <lb/><lb/></s></p></div> |
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3464 <head xml:id="echoid-head96" xml:space="preserve"> |
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|
3465 prop. 10. Supplementi |
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3466 <lb/>[<emph style="it">tr: |
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|
3467 Proposition 10 from the Supplementum |
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3468 </emph>]<lb/> |
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3469 </head> |
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3470 <p xml:lang="lat"> |
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3471 <s xml:id="echoid-s245" xml:space="preserve"> |
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3472 Si fuerint tres lineæ rectæ proportionales: Est ut prima ad tertiam, ita adgregatum <lb/> |
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|
3473 quadratorum primæ et secundæ ad adgregatum quadratorum secundæ et tertiæ. |
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3474 <lb/>[<emph style="it">tr: |
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3475 If there are three proportional lines, as the first is to the third, |
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3476 so is the sum of squares of the first and second to the sum of squares of the second and third. |
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3477 </emph>]<lb/> |
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3478 </s> |
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|
3479 <lb/> |
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3480 <s xml:id="echoid-s246" xml:space="preserve"> |
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3481 sint tres proportionales <lb/> |
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3482 continue <lb/> |
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3483 consequetur <lb/> |
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3484 vel <lb/> |
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3485 Et per synæresin <lb/> |
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3486 Et per 20,6 Euclid <lb/> |
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3487 Ergo pro conclusione |
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3488 <lb/>[<emph style="it">tr: |
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3489 let there be three continued proportionals <lb/> |
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3490 consequently <lb/> |
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3491 or <lb/> |
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3492 And by synæresis <lb/> |
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3493 And by Euclid VI.20 <lb/> |
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3494 Therefore in conclusion |
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3495 </emph>]<lb/> |
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3496 </s> |
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3497 </p> |
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3498 <head xml:id="echoid-head97" xml:space="preserve"> |
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3499 prop. 11. |
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3500 <lb/>[<emph style="it">tr: |
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3501 Proposition 11 |
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3502 </emph>]<lb/> |
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3503 </head> |
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3504 <p xml:lang="lat"> |
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3505 <s xml:id="echoid-s247" xml:space="preserve"> |
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3506 Si fuerint tres lineæ rectæ proportionales, est ut prima ad adgregatum primae et <lb/> |
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3507 tertiæ, ita quadratum secundæ ad adgregatum quadratorum secundæ et tertiæ. |
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3508 <lb/>[<emph style="it">tr: |
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3509 If there are three proportional lines, as the first is to the sum of the first and third, |
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3510 so is the square of the second to the sum of squares of the second and third. |
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3511 </emph>]<lb/> |
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3512 </s> |
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3513 <lb/> |
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3514 <s xml:id="echoid-s248" xml:space="preserve"> |
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3515 sint tres proportionales <lb/> |
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3516 per 20,6 El <lb/> |
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3517 Et per Synæresin <lb/> |
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3518 Concluditur |
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3519 <lb/>[<emph style="it">tr: |
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3520 let there be three proportionals <lb/> |
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3521 by Elements VI.20 <lb/> |
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3522 And by synæresin <lb/> |
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3523 It may be concluded. |
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3524 </emph>]<lb/> |
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3525 </s> |
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3526 </p> |
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3527 <head xml:id="echoid-head98" xml:space="preserve"> |
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3528 Consectarium |
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3529 <lb/>[<emph style="it">tr: |
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3530 Consequence |
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3531 </emph>]<lb/> |
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3532 </head> |
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3533 <p xml:lang="lat"> |
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3534 <s xml:id="echoid-s249" xml:space="preserve"> |
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3535 Itaque si fuerint tres lineæ rectæ proportionales, tria solida ab ijs <lb/> |
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|
3536 effecta æqualia sunt. |
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3537 per 10<emph style="super">am</emph> conculsionem <lb/> |
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3538 per 11<emph style="super">am</emph> conclu. <lb/> |
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3539 <lb/>[...]<lb/> <lb/> |
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3540 Dua prima solida sunt æqualia, quia unum factum est ab extremis analogia 10<emph style="super">am</emph> <lb/> |
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3541 et alterum a modijs. |
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3542 Tertium est factum a modijs <emph style="st">inferioris</emph> analogia 11<emph style="super">am</emph>, <lb/> |
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3543 cuius extremæ sunt eædem <emph style="st">superioris</emph> <emph style="super">analogia 10am</emph>, |
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3544 et illo æquale. |
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3545 <lb/>[<emph style="it">tr: |
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3546 Therefore if there are three lines in proportion, three solids constructed from them are equal. <lb/> |
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3547 by the conclusion of the 10th <lb/> |
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3548 by the conclusion of the 11th <lb/> |
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3549 <lb/>[...]<lb/> <lb/> |
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3550 The two first solids are equal, because one is made from the extremes of the ratio of the 10th, |
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3551 and the other by the method <lb/> |
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3552 The third is made by the method of the ratio of the 11th, whose extremes are the same as in the ratio of the 10th, |
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3553 and is equal to that one. |
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3554 </emph>]<lb/> |
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3555 </s> |
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3556 </p> |
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|
3557 <pb file="add_6784_f354v" o="354v" n="708"/> |
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3558 <pb file="add_6784_f355" o="355" n="709"/> |
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3559 <div xml:id="echoid-div54" type="page_commentary" level="2" n="54"> |
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3560 <p> |
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3561 <s xml:id="echoid-s250" xml:space="preserve">[<emph style="it">Note: |
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3562 <p> |
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3563 <s xml:id="echoid-s250" xml:space="preserve"> |
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|
3564 On this page Harriot examines a particular case arising from Proposition VII of Viète's |
|
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3565 <emph style="it">Supplementum geometriæ</emph> (1593), when the fourth proportional is twice the first. |
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|
3566 The same proposition is the subject of Chapter V of Viète's |
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|
3567 <emph style="it">Variorum responsorum libri VIII</emph>, which was also published in 1593. |
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3568 </s> |
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3569 <lb/> |
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3570 <quote xml:lang="lat"> |
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3571 Caput V <lb/> |
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3572 Propositio <lb/> |
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3573 Describere quatuor lineas rectas continue proportionales, quarum extremæ sint in ratione dupla. |
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3574 </quote> |
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3575 <lb/> |
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3576 <quote> |
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3577 Construct four lines in continued proportion, whose extremes are in double ratio. |
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3578 </quote> |
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3579 <lb/> |
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3580 <s xml:id="echoid-s251" xml:space="preserve"> |
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3581 The text in the <emph style="it">Variorum</emph> refers to the <emph style="it">Supplementum</emph>, |
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3582 indicating that the <emph style="it">Supplementum</emph> was written first. |
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3583 </s> |
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3584 </p> |
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3585 </emph>] |
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3586 <lb/><lb/></s></p></div> |
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3587 <head xml:id="echoid-head99" xml:space="preserve"> |
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3588 Ad Corollorium prop. 7. Supplementi. Et ad cap. 5. Resp. lib. 8. pag. 4. |
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3589 <lb/>[<emph style="it">tr: |
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3590 On a corollary to Proposition 7 of the Supplement. |
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3591 Also Chapter 5, Variorum liber responsorum, page 4. |
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3592 </emph>]<lb/> |
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3593 </head> |
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3594 <p xml:lang="lat"> |
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3595 <s xml:id="echoid-s253" xml:space="preserve"> |
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3596 Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> prima proportionalium, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> ea <lb/> |
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3597 cuius quadratum est triplum quadrati <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>. <lb/> |
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3598 Tum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> est dupla ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>; et per assumptum <lb/> |
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3599 ex poristicis in alia charta demonstratum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> <lb/> |
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3600 erit quarta proportionalis. Per propositione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>A</mi></mstyle></math> est secunda et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>G</mi></mstyle></math> tertia. <lb/> |
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3601 Sed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>B</mi></mstyle></math> est æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>G</mi></mstyle></math> propter similitudine triangulorum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>F</mi><mi>B</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>A</mi><mi>C</mi></mstyle></math>, et <lb/> |
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3602 analogiam precedentam ut sequitur. <lb/> |
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3603 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mo>.</mo><mi>E</mi><mi>A</mi><mo>.</mo><mi>E</mi><mi>G</mi><mo>.</mo><mi>A</mi><mi>C</mi><mo>.</mo></mstyle></math> Analogia precedens. <lb/> |
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3604 <lb/>[...]<lb/> <lb/> |
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3605 Et per similitudi-<lb/> |
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3606 num Δ<emph style="super">orum</emph>. |
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3607 <lb/>[...]<lb/> <lb/> |
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3608 Ergo. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mo>.</mo><mi>A</mi><mi>E</mi><mo>.</mo><mi>F</mi><mi>B</mi><mo>.</mo><mi>A</mi><mi>C</mi><mo>.</mo></mstyle></math> continue proportionales. |
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3609 <lb/>[<emph style="it">tr: |
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3610 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> be the first proportional, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> that whose square is three times the square of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>. <lb/> |
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3611 Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> is twice <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>; and by taking it from the proof demonstrated in the other sheet, |
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3612 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> will be the fourth proportional. By the proposition <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>A</mi></mstyle></math> is the second and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>G</mi></mstyle></math> the third. <lb/> |
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3613 But <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>B</mi></mstyle></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>G</mi></mstyle></math> because of similar triangles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>F</mi><mi>B</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>A</mi><mi>C</mi></mstyle></math>, and <lb/> |
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3614 the precding ratio, as follows. <lb/> |
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3615 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mo>:</mo><mi>E</mi><mi>A</mi><mo>:</mo><mi>E</mi><mi>G</mi><mo>:</mo><mi>A</mi><mi>C</mi></mstyle></math> preceding ratio. <lb/> |
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3616 <lb/>[...]<lb/> <lb/> |
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3617 And by similar triangles. <lb/> |
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3618 <lb/>[...]<lb/> <lb/> |
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3619 Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mo>:</mo><mi>A</mi><mi>E</mi><mo>:</mo><mi>F</mi><mi>B</mi><mo>:</mo><mi>A</mi><mi>C</mi></mstyle></math> are continued proportionals. |
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3620 </emph>]<lb/> |
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3621 [<emph style="it">Note: |
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3622 The other sheet mentioned in this paragraph appears to be Add MS 6784, f. 356. |
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3623 </emph>]<lb/> |
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3624 </s> |
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3625 </p> |
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3626 <p xml:lang="lat"> |
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3627 <s xml:id="echoid-s254" xml:space="preserve"> |
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3628 Datis igitur extremis in ratione dupla, mediæ ita compendiosæ <lb/> |
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3629 inveniuntur. |
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3630 <lb/>[<emph style="it">tr: |
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3631 Therefore given the extremes in double ratio, the mean is briefly found. |
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3632 </emph>]<lb/> |
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3633 </s> |
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3634 <lb/> |
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3635 <s xml:id="echoid-s255" xml:space="preserve"> |
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3636 Sit maxima <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> bisariam divisa in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> et intervallo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> describatur <lb/> |
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3637 circulus. Et sit <emph style="st">prima</emph> <emph style="super">minima</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> inscripta |
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3638 et producta ad partes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math>. <lb/> |
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3639 Ducatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>E</mi></mstyle></math> ita ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>F</mi></mstyle></math> sit æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>. et acta fit linea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>B</mi></mstyle></math>. <lb/> |
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3640 Quatuor igitur continue proportionales ex supra demonstratis sunt. |
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3641 <lb/>[<emph style="it">tr: |
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3642 Let the maximum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> be cut in half at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> and with radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> there is described a circle. |
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3643 And let the minimum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> be inscribed and produced to the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math>. |
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3644 Construct <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>E</mi></mstyle></math> so that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>F</mi></mstyle></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>, and let the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>B</mi></mstyle></math> be joined. <lb/> |
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3645 Therefore there are the four continued proportionals that were demonstrated above. |
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3646 </emph>]<lb/> |
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3647 </s> |
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3648 </p> |
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3649 <pb file="add_6784_f355v" o="355v" n="710"/> |
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3650 <pb file="add_6784_f356" o="356" n="711"/> |
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3651 <div xml:id="echoid-div55" type="page_commentary" level="2" n="55"> |
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3652 <p> |
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3653 <s xml:id="echoid-s256" xml:space="preserve">[<emph style="it">Note: |
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3654 <p> |
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3655 <s xml:id="echoid-s256" xml:space="preserve"> |
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3656 On this page Harriot examines a particular case arising from Proposition VII of Viète's |
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3657 <emph style="it">Supplementum geometriæ</emph> (1593), when the fourth proportional is twice the first. |
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3658 </s> |
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3659 </p> |
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3660 </emph>] |
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3661 <lb/><lb/></s></p></div> |
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3662 <head xml:id="echoid-head100" xml:space="preserve"> |
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3663 prop. 7. Supplementi de corrollario |
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3664 <lb/>[<emph style="it">tr: |
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3665 Proposition 7 of the Supplement, on a corollary |
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3666 </emph>]<lb/> |
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3667 </head> |
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3668 <p xml:lang="lat"> |
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3669 <s xml:id="echoid-s258" xml:space="preserve"> |
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3670 Sint 4<emph style="super">or</emph> proportionales <lb/> |
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3671 in specie. <lb/> |
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3672 Si quarta sit dupla ad prima, erit: <lb/> |
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3673 <lb/>[...]<lb/> <lb/> |
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3674 Ergo quatuor proportionales <lb/> |
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3675 quarum extremæ sunt in <lb/> |
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3676 ratione dupla erunt |
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3677 <lb/>[<emph style="it">tr: |
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3678 Let there be 4 proportionals in general form. <lb/> |
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3679 If the fourth is twice the firs, then: <lb/> |
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3680 <lb/>[...]<lb/> <lb/> |
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3681 Therefore the four proportionals whose extremes are in double ratio will be |
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3682 </emph>]<lb/> |
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3683 </s> |
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3684 </p> |
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3685 <p xml:lang="lat"> |
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3686 <s xml:id="echoid-s259" xml:space="preserve"> |
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3687 Tunc fac <lb/>[...]<lb/> et nota quadratorum differentiam. |
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3688 <lb/>[<emph style="it">tr: |
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3689 Then make [the square of the first and second and the square of the third and fourth], |
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3690 and note the difference of the squares.</emph>]<lb/> |
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3691 </s> |
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3692 </p> |
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3693 <p xml:lang="lat"> |
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3694 <s xml:id="echoid-s260" xml:space="preserve"> |
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3695 Differentia quadratorum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi></mstyle></math><lb/> |
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3696 Hoc est triplum quadratum primæ proportionalis. |
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3697 <lb/>[<emph style="it">tr: |
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3698 The difference of the squares is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>b</mi><mi>b</mi></mstyle></math>. <lb/> |
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3699 This is three times the square of the first proportional. |
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3700 </emph>]<lb/> |
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3701 </s> |
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3702 </p> |
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3703 <pb file="add_6784_f356v" o="356v" n="712"/> |
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3704 <pb file="add_6784_f357" o="357" n="713"/> |
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3705 <div xml:id="echoid-div56" type="page_commentary" level="2" n="56"> |
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3706 <p> |
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3707 <s xml:id="echoid-s261" xml:space="preserve">[<emph style="it">Note: |
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3708 <p> |
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3709 <s xml:id="echoid-s261" xml:space="preserve"> |
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3710 This page investigates the proposition that is the subject of Chapter V of Viète's |
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3711 <emph style="it">Variorum responsorum libri VIII</emph>. |
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3712 It appears to be a continuation of Add MS 6784, f. 355. |
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3713 </s> |
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3714 <lb/> |
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3715 <quote xml:lang="lat"> |
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3716 Caput V <lb/> |
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3717 Propositio <lb/> |
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3718 Describere quatuor lineas rectas continue proportionales, quarum extremæ sint in ratione dupla. |
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3719 </quote> |
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3720 <lb/> |
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|
3721 <quote> |
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3722 Construct four lines in continued proportion, whose extremes are in double ratio. |
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3723 </quote> |
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3724 </p> |
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|
3725 </emph>] |
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3726 <lb/><lb/></s></p></div> |
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|
3727 <head xml:id="echoid-head101" xml:space="preserve"> |
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|
3728 In Cap. 5. Resp. lib. 8. pag. 4. |
|
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3729 <lb/>[<emph style="it">tr: |
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|
3730 Chapter 5, Variorum liber responsorum, page 4. |
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3731 </emph>]<lb/> |
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3732 </head> |
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3733 <pb file="add_6784_f357v" o="357v" n="714"/> |
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3734 <pb file="add_6784_f358" o="358" n="715"/> |
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3735 <div xml:id="echoid-div57" type="page_commentary" level="2" n="57"> |
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3736 <p> |
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3737 <s xml:id="echoid-s263" xml:space="preserve">[<emph style="it">Note: |
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3738 <p> |
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3739 <s xml:id="echoid-s263" xml:space="preserve"> |
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|
3740 On this page Harriot examines Proposition VII from Viète's |
|
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3741 <emph style="it">Supplementum geometriæ</emph> (1593). |
|
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3742 </s> |
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3743 <lb/> |
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3744 <quote xml:lang="lat"> |
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|
3745 Propositio VII. <lb/> |
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3746 Data è tribus propositis lineis rectis proportionalibus prima, |
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3747 & ea cujus quadratum æquale fit ei quo differt quadratum compositae ex secunda & tertia |
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3748 àquadrato compositæ ex secunda & prima, invenire secundam & tertiam proprtionales. |
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|
3749 </quote> |
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3750 <lb/> |
|
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|
3751 <quote> |
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|
3752 Given the first of three proposed proportional straight lines, |
|
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|
3753 and another whose square is equal to the difference between the square of the sum of the second and third, |
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|
3754 and the square of the sum of the second and first, find the second and third proportionals. |
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|
3755 </quote> |
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|
3756 </p> |
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|
3757 </emph>] |
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|
3758 <lb/><lb/></s></p></div> |
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|
3759 <head xml:id="echoid-head102" xml:space="preserve"> |
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|
3760 prop. 7. Supplementi |
|
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3761 <lb/>[<emph style="it">tr: |
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|
3762 Proposition 7 of the Supplement |
|
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|
3763 </emph>]<lb/> |
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|
3764 </head> |
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|
3765 <p xml:lang="lat"> |
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3766 <s xml:id="echoid-s265" xml:space="preserve"> |
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|
3767 Data e tribus propositis lineis rectis proportionalibus prima et ea <lb/> |
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|
3768 cujus quadratum aequale fit ei quo differt quadratum compositae ex <lb/> |
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|
3769 secunda et tertia a quadrato compositæ ex secunda et prima: invenire <lb/> |
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|
3770 secundam et tertiam proprtionales. |
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|
3771 <lb/>[<emph style="it">tr: |
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|
3772 Given the first of three proposed proportional straight lines, |
|
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|
3773 and another whose square is equal to the difference between the square of the sum of the second and third, |
|
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|
3774 and the square of the sum of the second and first, find the second and third proportionals. |
|
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|
3775 </emph>]<lb/> |
|
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|
3776 </s> |
|
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|
3777 </p> |
|
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|
3778 <p xml:lang="lat"> |
|
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|
3779 <s xml:id="echoid-s266" xml:space="preserve"> |
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|
3780 Data prima <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> <lb/> |
|
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|
3781 Et recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> |
|
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|
3782 <lb/>[<emph style="it">tr: |
|
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|
3783 The first given line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> and the straight line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math>. |
|
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|
3784 </emph>]<lb/> |
|
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|
3785 </s> |
|
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|
3786 </p> |
|
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|
3787 <p xml:lang="lat"> |
|
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|
3788 <s xml:id="echoid-s267" xml:space="preserve"> |
|
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|
3789 Tum tres proportionales <lb/> |
|
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|
3790 erunt. |
|
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|
3791 <lb/>[<emph style="it">tr: |
|
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|
3792 Then the three proportionals will be: |
|
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|
3793 </emph>]<lb/> |
|
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|
3794 </s> |
|
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|
3795 </p> |
|
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|
3796 <pb file="add_6784_f358v" o="358v" n="716"/> |
|
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|
3797 <pb file="add_6784_f359" o="359" n="717"/> |
|
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|
3798 <head xml:id="echoid-head103" xml:space="preserve"> |
|
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|
3799 a) Achilles |
|
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|
3800 </head> |
|
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|
3801 <p xml:lang="lat"> |
|
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|
3802 <s xml:id="echoid-s268" xml:space="preserve"> |
|
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|
3803 Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, Achilles. <lb/> |
|
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|
3804 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>, testudo. |
|
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|
3805 <lb/>[<emph style="it">tr: |
|
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|
3806 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be Achilles, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math> the tortoise. |
|
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3807 </emph>]<lb/> |
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3808 </s> |
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3809 </p> |
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3810 <p xml:lang="lat"> |
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3811 <s xml:id="echoid-s269" xml:space="preserve"> |
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3812 Sit ratio motus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, ad motus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>, <lb/> |
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3813 ut: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. <lb/> |
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3814 nempe: 10 ad 1. |
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3815 <lb/>[<emph style="it">tr: |
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3816 Let the ratio of the motion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> to the motion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math> be as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, namely, 1 to 10. |
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3817 </emph>]<lb/> |
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3818 </s> |
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3819 </p> |
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3820 <p xml:lang="lat"> |
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3821 <s xml:id="echoid-s270" xml:space="preserve"> |
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3822 Et sit distantia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. nempe 1 <foreign xml:lang="fr">mille</foreign> pases. |
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3823 <lb/>[<emph style="it">tr: |
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3824 And let the distance between <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, namely, one thousand pases. |
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3825 </emph>]<lb/> |
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3826 </s> |
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3827 </p> |
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3828 <p xml:lang="lat"> |
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3829 <s xml:id="echoid-s271" xml:space="preserve"> |
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3830 Et sit motus utriusque in eadem linea et ad easdem partes, nempe <lb/> |
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3831 ab <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math> versus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>. |
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3832 <lb/>[<emph style="it">tr: |
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3833 And suppose the motion of both is in the same line and in the same direction, |
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3834 namely, from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math> towards <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>. |
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3835 </emph>]<lb/> |
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3836 </s> |
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3837 </p> |
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3838 <p xml:lang="lat"> |
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3839 <s xml:id="echoid-s272" xml:space="preserve"> |
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3840 Quæritur ex datis punctum ubi Achilles comprehendet testudinem. |
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3841 <lb/>[<emph style="it">tr: |
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3842 From what is given there is sought the point where Achilles catches up with the tortoise. |
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3843 </emph>]<lb/> |
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3844 </s> |
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3845 </p> |
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3846 <p xml:lang="lat"> |
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3847 <s xml:id="echoid-s273" xml:space="preserve"> |
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3848 Quæestio solvitur exhibendo summam infinitæ progressionis decrescentis <lb/> |
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3849 ut sequitur: (species summa infinitæ progressionis decrescentis <lb/> |
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3850 ut in doctrinam de <reg norm="progressionis" type="abbr">prog</reg>: |
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3851 <reg norm="geometricæ" type="abbr">geom</reg>: est:) |
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3852 <lb/>[<emph style="it">tr: |
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3853 The problem is solved by producing the sum of an infinite decreasing progression as follows: |
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3854 (the case of the sum of an infinite decreasing progression as in the teaching of geometric porgressions is:) |
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3855 </emph>]<lb/> |
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3856 </s> |
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3857 </p> |
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3858 <p xml:lang="lat"> |
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3859 <s xml:id="echoid-s274" xml:space="preserve"> |
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|
3860 Alia progressiones. |
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3861 <lb/>[<emph style="it">tr: |
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3862 Other progressions. |
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3863 </emph>]<lb/> |
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|
3864 </s> |
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|
3865 </p> |
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3866 <p xml:lang="lat"> |
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3867 <s xml:id="echoid-s275" xml:space="preserve"> |
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|
3868 (ut Archimedes de <lb/> |
|
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|
3869 quad: parab: pr: 23) |
|
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|
3870 <lb/>[<emph style="it">tr: |
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|
3871 (as Archimedes in the quadrature of the parabola, proposition 23) |
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|
3872 </emph>]<lb/> |
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|
3873 </s> |
|
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|
3874 </p> |
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|
3875 <pb file="add_6784_f359v" o="359v" n="718"/> |
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|
3876 <pb file="add_6784_f360" o="360" n="719"/> |
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|
3877 <head xml:id="echoid-head104" xml:space="preserve"> |
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|
3878 b) Achilles |
|
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|
3879 </head> |
|
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|
3880 <p xml:lang="lat"> |
|
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|
3881 <s xml:id="echoid-s276" xml:space="preserve"> |
|
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|
3882 Sit (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>), Achilles. <lb/> |
|
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3883 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>, testudo. |
|
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|
3884 <lb/>[<emph style="it">tr: |
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|
3885 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be Achilles, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math> the tortoise. |
|
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3886 </emph>]<lb/> |
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3887 </s> |
|
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3888 </p> |
|
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3889 <p xml:lang="lat"> |
|
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|
3890 <s xml:id="echoid-s277" xml:space="preserve"> |
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|
3891 Sit velocitas motus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, ad velocitatem motus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>, <lb/> |
|
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|
3892 ut: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. |
|
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|
3893 <lb/>[<emph style="it">tr: |
|
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|
3894 Let the speed of motion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> to the speed of motion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math> be as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. |
|
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|
3895 </emph>]<lb/> |
|
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|
3896 </s> |
|
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|
3897 </p> |
|
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|
3898 <p xml:lang="lat"> |
|
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|
3899 <s xml:id="echoid-s278" xml:space="preserve"> |
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|
3900 Sit distantia inter (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>) et (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>). <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. |
|
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|
3901 <lb/>[<emph style="it">tr: |
|
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|
3902 Let the distance between <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. |
|
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|
3903 </emph>]<lb/> |
|
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|
3904 </s> |
|
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|
3905 </p> |
|
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|
3906 <p xml:lang="lat"> |
|
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|
3907 <s xml:id="echoid-s279" xml:space="preserve"> |
|
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|
3908 Et sit motus utriusque in eadem linea et ad easdem partes, nempe ab (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>), et (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>) <lb/> |
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|
3909 versus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>. |
|
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|
3910 <lb/>[<emph style="it">tr: |
|
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|
3911 And let the mtion of both be in the same line and the same direction, |
|
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|
3912 namely from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math> towards <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>. |
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|
3913 </emph>]<lb/> |
|
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|
3914 </s> |
|
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|
3915 </p> |
|
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|
3916 <p xml:lang="lat"> |
|
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|
3917 <s xml:id="echoid-s280" xml:space="preserve"> |
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|
3918 Quæritur ex datis punctum ubi Achilles comprehendet testudinem. |
|
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|
3919 <lb/>[<emph style="it">tr: |
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|
3920 From what is given there is sought the point where Achilles catches up with the tortoise. |
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|
3921 </emph>]<lb/> |
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|
3922 </s> |
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|
3923 </p> |
|
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|
3924 <p xml:lang="lat"> |
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|
3925 <s xml:id="echoid-s281" xml:space="preserve"> |
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|
3926 Ponatur illud punctum esse <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>. et sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi><mi>w</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. |
|
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|
3927 <lb/>[<emph style="it">tr: |
|
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|
3928 Suppose this point is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>, and let the distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi><mi>w</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. |
|
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|
3929 </emph>]<lb/> |
|
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|
3930 </s> |
|
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|
3931 <lb/> |
|
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|
3932 <s xml:id="echoid-s282" xml:space="preserve"> |
|
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|
3933 Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. Et inde <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>. |
|
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|
3934 <lb/>[<emph style="it">tr: |
|
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|
3935 Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> is found; and hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>. |
|
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|
3936 </emph>]<lb/> |
|
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|
3937 </s> |
|
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|
3938 </p> |
|
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|
3939 <p xml:lang="lat"> |
|
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|
3940 <s xml:id="echoid-s283" xml:space="preserve"> |
|
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|
3941 In numeris sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. 10. <lb/> |
|
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|
3942 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. 2. <lb/> |
|
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|
3943 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. 2. mill |
|
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|
3944 <lb/>[<emph style="it">tr: |
|
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|
3945 In numbers let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>1</mn><mn>0</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>2</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mn>2</mn></mstyle></math> miles |
|
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|
3946 </emph>]<lb/> |
|
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|
3947 </s> |
|
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|
3948 </p> |
|
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|
3949 <p xml:lang="lat"> |
|
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|
3950 <s xml:id="echoid-s284" xml:space="preserve"> |
|
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|
3951 Aliter. |
|
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|
3952 <lb/>[<emph style="it">tr: |
|
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|
3953 Another way. |
|
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|
3954 </emph>]<lb/> |
|
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|
3955 </s> |
|
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|
3956 </p> |
|
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|
3957 <p xml:lang="lat"> |
|
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|
3958 <s xml:id="echoid-s285" xml:space="preserve"> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
3959 Aliter 2<emph style="super">o</emph>. <lb/> |
|
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|
3960 Quæritur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>w</mi></mstyle></math> et sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. |
|
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changeset
|
3961 <lb/>[<emph style="it">tr: |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
3962 A second way. <lb/> |
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3963 There is sought <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>w</mi></mstyle></math>, and suppose it is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. |
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3964 </emph>]<lb/> |
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3965 </s> |
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3966 </p> |
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3967 <p xml:lang="lat"> |
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3968 <s xml:id="echoid-s286" xml:space="preserve"> |
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3969 Exemplum de duabus [¿]numeribus[?]. |
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3970 <lb/>[<emph style="it">tr: |
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3971 An example from two numbers. |
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3972 </emph>]<lb/> |
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3973 </s> |
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3974 </p> |
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3975 <pb file="add_6784_f360v" o="360v" n="720"/> |
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3976 <pb file="add_6784_f361" o="361" n="721"/> |
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3977 <div xml:id="echoid-div58" type="page_commentary" level="2" n="58"> |
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3978 <p> |
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3979 <s xml:id="echoid-s287" xml:space="preserve">[<emph style="it">Note: |
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3980 <p> |
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3981 <s xml:id="echoid-s287" xml:space="preserve"> |
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3982 On this folio, Harriot derives the sum of a finite geometric progression, |
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3983 using Euclid V.12 and its numerical counterpoart, Euclid VII.12. |
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3984 He then extends his result to an infinite (decreasing) progression, |
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3985 by arguing that the final term must be infnitely small, that is, nothing. <lb/> |
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3986 Euclid V.12: If any number of magnitudes be proportional, |
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3987 as one of the antecedents is to one of the consequents, |
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3988 so will all the antecedents be to all the consequents. <lb/> |
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3989 Euclid VII.12: If there be as many numbers as we please in proportion, then, |
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3990 as one of the antecedents is to one of the consequents, |
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3991 so are all the antecedents to all the consequents. |
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3992 </s> |
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3993 </p> |
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3994 </emph>] |
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3995 <lb/><lb/></s></p></div> |
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3996 <head xml:id="echoid-head105" xml:space="preserve" xml:lang="lat"> |
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3997 1.) De progressione geometrica. |
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3998 <lb/>[<emph style="it">tr: |
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3999 On geometric porgressions |
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4000 </emph>]<lb/> |
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4001 </head> |
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4002 <p xml:lang="lat"> |
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4003 <s xml:id="echoid-s289" xml:space="preserve"> |
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4004 Theorema. |
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4005 <lb/>[<emph style="it">tr: |
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4006 Theorem |
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4007 </emph>]<lb/> |
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4008 </s> |
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4009 <lb/> |
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4010 <s xml:id="echoid-s290" xml:space="preserve"> |
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4011 el. 5. pr: 12. |
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4012 <lb/>[<emph style="it">tr: |
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4013 <emph style="it">Elements</emph>, Book 5, Proposition 12. |
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4014 </emph>]<lb/> |
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4015 </s> |
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4016 <lb/> |
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4017 <s xml:id="echoid-s291" xml:space="preserve"> |
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4018 el. 7. pr. 12. |
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4019 <lb/>[<emph style="it">tr: |
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4020 <emph style="it">Elements</emph>, Book 7, Proposition 12. |
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4021 </emph>]<lb/> |
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4022 </s> |
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4023 <lb/> |
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4024 <s xml:id="echoid-s292" xml:space="preserve"> |
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4025 Si sint magnitudines quotcunque proportionales, Quemadmodum <lb/> |
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4026 se habuerit una antecedentium ad unam consequentium: Ita <lb/> |
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4027 se habebunt omnes antecedentes ad omnes consequentes. |
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4028 <lb/>[<emph style="it">tr: |
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4029 If any number of magnitudes are proportional, |
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4030 then just as as one antecedent is to its consequent, |
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4031 so will the sum of the antecedents be to the sum of the consequents. |
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4032 </emph>]<lb/> |
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4033 </s> |
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4034 </p> |
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4035 <p xml:lang="lat"> |
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4036 <s xml:id="echoid-s293" xml:space="preserve"> |
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4037 Sint continue proportionales. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. |
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4038 <lb/>[<emph style="it">tr: |
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4039 Let the continued proportionals be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. |
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4040 </emph>]<lb/> |
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4041 </s> |
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4042 </p> |
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4043 <p xml:lang="lat"> |
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4044 <s xml:id="echoid-s294" xml:space="preserve"> |
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4045 In notis universalibus sit. |
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4046 <lb/>[<emph style="it">tr: |
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4047 In general notation we have |
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4048 </emph>]<lb/> |
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4049 </s> |
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4050 <lb/> |
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4051 <s xml:id="echoid-s295" xml:space="preserve"> |
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4052 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. primum. <emph style="st">p</emph>. primus terminus rationis. |
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4053 <lb/>[<emph style="it">tr: |
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4054 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. first term. <emph style="st">p</emph>. first term of the ratio. |
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4055 </emph>]<lb/> |
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4056 </s> |
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4057 <lb/> |
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4058 <s xml:id="echoid-s296" xml:space="preserve"> |
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4059 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>. secunda. <emph style="st">s</emph>. secundus. |
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4060 <lb/>[<emph style="it">tr: |
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4061 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>. second. <emph style="st">s</emph>. second. |
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4062 </emph>]<lb/> |
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4063 </s> |
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4064 <lb/> |
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4065 <s xml:id="echoid-s297" xml:space="preserve"> |
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4066 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. ultima. |
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4067 <lb/>[<emph style="it">tr: |
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4068 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. last. |
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4069 </emph>]<lb/> |
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4070 </s> |
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4071 <lb/> |
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4072 <s xml:id="echoid-s298" xml:space="preserve"> |
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4073 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. omnes. |
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4074 <lb/>[<emph style="it">tr: |
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4075 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. all. |
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4076 </emph>]<lb/> |
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4077 </s> |
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4078 </p> |
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4079 <p xml:lang="lat"> |
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4080 <s xml:id="echoid-s299" xml:space="preserve"> |
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4081 Ergo; si, <emph style="st">p</emph> > <emph style="st">s</emph> ut in progressi decrescente: |
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4082 <lb/>[<emph style="it">tr: |
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4083 Therfore if <emph style="st">p</emph> > <emph style="st">s</emph> are in a decreasing progression: |
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4084 </emph>]<lb/> |
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4085 </s> |
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4086 </p> |
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4087 <p xml:lang="lat"> |
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4088 <s xml:id="echoid-s300" xml:space="preserve"> |
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4089 Ergo; si, <emph style="st">p</emph> < <emph style="st">s</emph> ut in progressi crescente: |
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4090 <lb/>[<emph style="it">tr: |
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4091 Therfore if <emph style="st">p</emph> > <emph style="st">s</emph> are in an increasing progression: |
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4092 </emph>]<lb/> |
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4093 </s> |
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4094 </p> |
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4095 <p xml:lang="lat"> |
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4096 <s xml:id="echoid-s301" xml:space="preserve"> |
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4097 De <emph style="st">infinitis</emph> progressionibus <lb/> |
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4098 decrescentibus in infinitum: |
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4099 <lb/>[<emph style="it">tr: |
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4100 For a progression descreasing indefinitely: |
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4101 </emph>]<lb/> |
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4102 </s> |
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4103 <lb/> |
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4104 <s xml:id="echoid-s302" xml:space="preserve"> |
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4105 Cum progressio decrescit et <lb/> |
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4106 numerus terminorum sit infinitus; <lb/> |
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4107 ultimus terminus est infinite <lb/> |
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4108 minimus hoc est nullius quantiatis. |
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4109 <lb/>[<emph style="it">tr: |
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4110 Since the progression decreases and the number of terms is infinite, the last term is infnitely small, |
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4111 that is, of no quantity. |
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4112 </emph>]<lb/> |
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4113 </s> |
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4114 <lb/> |
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4115 <s xml:id="echoid-s303" xml:space="preserve"> |
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4116 Ideo: |
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4117 <lb/>[<emph style="it">tr: |
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4118 Therefore. |
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4119 </emph>]<lb/> |
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4120 </s> |
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4121 </p> |
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4122 <pb file="add_6784_f361v" o="361v" n="722"/> |
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4123 <pb file="add_6784_f362" o="362" n="723"/> |
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4124 <div xml:id="echoid-div59" type="page_commentary" level="2" n="59"> |
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4125 <p> |
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4126 <s xml:id="echoid-s304" xml:space="preserve">[<emph style="it">Note: |
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4127 <p> |
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4128 <s xml:id="echoid-s304" xml:space="preserve"> |
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|
4129 In the preceding folio, f. 361, Harriot derived a formula for the sum of a finite geometric progression |
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4130 based on Euclid V.12. Here he gives an alternative derivation based on Euclid IX. 35. <lb/> |
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4131 Euclid IX. 35: If as many numbers as we please be in continued proportion, |
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4132 and there be subtracted from the second and the last numbers equal to the first, |
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4133 then as the excess of the second is to the first, |
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4134 so will the excess of the last be to all those before it. |
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4135 </s> |
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4136 </p> |
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4137 </emph>] |
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4138 <lb/><lb/></s></p></div> |
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4139 <head xml:id="echoid-head106" xml:space="preserve" xml:lang="lat"> |
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4140 2.) De progressione geometrica. |
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4141 <lb/>[<emph style="it">tr: |
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4142 On geometric porgressions |
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4143 </emph>]<lb/> |
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4144 </head> |
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4145 <p xml:lang="lat"> |
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4146 <s xml:id="echoid-s306" xml:space="preserve"> |
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4147 Theoremata. |
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4148 <lb/>[<emph style="it">tr: |
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4149 Theorem |
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4150 </emph>]<lb/> |
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4151 </s> |
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4152 <lb/> |
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4153 <s xml:id="echoid-s307" xml:space="preserve"> |
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4154 el. 9. pr: 35. |
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4155 <lb/>[<emph style="it">tr: |
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4156 <emph style="it">Elements</emph> Book IX, Proposition 35 |
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4157 </emph>]<lb/> |
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4158 </s> |
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4159 <lb/> |
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4160 <s xml:id="echoid-s308" xml:space="preserve"> |
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4161 Si sint quotlibet numeri deinceps proportionales, detrahuntur autem <lb/> |
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4162 de secundo et ultimo æquales ipsi primo: erit quemadmodum <lb/> |
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4163 secundi excessus ad primum, ita ultima excessus ad omnes qui ultimum <lb/> |
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4164 antecedunt. |
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4165 <lb/>[<emph style="it">tr: |
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4166 If there are as many numbers as we please in proportion, |
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4167 and the first is subtracted from the second and the last, |
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4168 then just as the difference of the second is to the first, |
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4169 so is the difference of the last to all before the last. |
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4170 </emph>]<lb/> |
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4171 </s> |
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4172 </p> |
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4173 <p xml:lang="lat"> |
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4174 <s xml:id="echoid-s309" xml:space="preserve"> |
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4175 Progressio crescens: |
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4176 <lb/>[<emph style="it">tr: |
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4177 An increasing progression: |
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4178 </emph>]<lb/> |
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4179 </s> |
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4180 </p> |
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4181 <p xml:lang="lat"> |
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4182 <s xml:id="echoid-s310" xml:space="preserve"> |
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4183 In notis universalibus: sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, primus: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>, secundus: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, ultimus: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, omnes. |
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4184 <lb/>[<emph style="it">tr: |
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4185 In general notation, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> be the first term; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> the second term; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math> the last term; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> the sum. |
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4186 </emph>]<lb/> |
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4187 </s> |
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4188 </p> |
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4189 <p xml:lang="lat"> |
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4190 <s xml:id="echoid-s311" xml:space="preserve"> |
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4191 Progressio decrescens: |
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4192 <lb/>[<emph style="it">tr: |
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4193 A decreasing progression: |
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4194 </emph>]<lb/> |
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4195 </s> |
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4196 </p> |
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4197 <p xml:lang="lat"> |
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4198 <s xml:id="echoid-s312" xml:space="preserve"> |
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|
4199 In notis universalis erit: |
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4200 <lb/>[<emph style="it">tr: |
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|
4201 In general notation we have: |
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4202 </emph>]<lb/> |
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4203 </s> |
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4204 </p> |
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4205 <p xml:lang="lat"> |
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4206 <s xml:id="echoid-s313" xml:space="preserve"> |
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|
4207 Vel: in notis magis universalis. <lb/> |
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4208 sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, primus terminus rationis. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>, secundus. <lb/> |
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4209 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math>, maxumus terminus progressionis <lb/> |
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4210 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>, minimus. Tum: |
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4211 <lb/>[<emph style="it">tr: |
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4212 Or, in more general notation, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> be the first term of the ratio, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> the second, |
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4213 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math> the greatest term of the progression, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math> the least. Then: |
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4214 </emph>]<lb/> |
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4215 </s> |
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4216 </p> |
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4217 <pb file="add_6784_f362v" o="362v" n="724"/> |
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4218 <pb file="add_6784_f363" o="363" n="725"/> |
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4219 <div xml:id="echoid-div60" type="page_commentary" level="2" n="60"> |
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4220 <p> |
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4221 <s xml:id="echoid-s314" xml:space="preserve">[<emph style="it">Note: |
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4222 <p> |
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4223 <s xml:id="echoid-s314" xml:space="preserve"> |
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4224 In this folio Harriot repeats statements that are to be found in Viete, |
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4225 <emph style="it">Variorum responsorum</emph>, Chapter XVII (1646, 397–398). <lb/> |
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4226 Harriot's letters <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, <emph style="st">M</emph>, <emph style="st">m</emph> |
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4227 correspond to Viete's <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>X</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>. <lb/> |
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4228 Harriot's final comments refer to the final sentence of Viete's penultimate paragraph (1646, 398): <lb/> |
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4229 <foreign xml:lang="lat"> |
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4230 Et ut differentia terminorum rationis ad terminorum rationis majorem, |
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4231 ita maxima ad compositam ex ombnibus plus cremento. |
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4232 </foreign> <lb/> |
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4233 <lb/>[<emph style="it">tr: |
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4234 As the difference in the terms of the ratio is to the greater term of the ratio, |
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4235 so is the the greatest term of the progression to the sum plus an increment. |
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4236 </emph>]<lb/> |
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4237 </s> |
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4238 </p> |
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4239 </emph>] |
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4240 <lb/><lb/></s></p></div> |
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4241 <head xml:id="echoid-head107" xml:space="preserve" xml:lang="lat"> |
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4242 3.) De progressione geometrica. (ut Vieta in var: resp.) |
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4243 <lb/>[<emph style="it">tr: |
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4244 On geometric progressions (as Viete in <emph style="it">Variorum responsorum</emph>) |
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4245 </emph>]<lb/> |
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4246 </head> |
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4247 <p xml:lang="lat"> |
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4248 <s xml:id="echoid-s316" xml:space="preserve"> |
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4249 Crescente. |
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4250 <lb/>[<emph style="it">tr: |
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4251 Increasing. |
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4252 </emph>]<lb/> |
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4253 </s> |
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4254 <s xml:id="echoid-s317" xml:space="preserve"> |
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4255 decrescente. |
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4256 <lb/>[<emph style="it">tr: |
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4257 Decreasing. |
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4258 </emph>]<lb/> |
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4259 </s> |
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4260 </p> |
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4261 <p xml:lang="lat"> |
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4262 <s xml:id="echoid-s318" xml:space="preserve"> |
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4263 <emph style="st">m</emph>. minor terminus rationis. |
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4264 <lb/>[<emph style="it">tr: |
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4265 Let <emph style="st">m</emph> be the lesser terms of the ratio. |
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4266 </emph>]<lb/> |
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4267 </s> |
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4268 <lb/> |
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4269 <s xml:id="echoid-s319" xml:space="preserve"> |
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4270 <emph style="st">M</emph>. Maior terminus rationis. |
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4271 <lb/>[<emph style="it">tr: |
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4272 Let <emph style="st">M</emph> be the greater terms of the ratio. |
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4273 </emph>]<lb/> |
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4274 </s> |
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4275 <lb/> |
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4276 <s xml:id="echoid-s320" xml:space="preserve"> |
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4277 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math>. maximus terminus progressionis. |
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4278 <lb/>[<emph style="it">tr: |
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4279 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math> be the greatest term of the progression. |
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4280 </emph>]<lb/> |
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4281 </s> |
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4282 <lb/> |
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4283 <s xml:id="echoid-s321" xml:space="preserve"> |
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4284 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>. minimus terminus progressionis. |
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4285 <lb/>[<emph style="it">tr: |
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4286 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math> be the least term of the progression. |
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4287 </emph>]<lb/> |
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4288 </s> |
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4289 <lb/> |
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4290 <s xml:id="echoid-s322" xml:space="preserve"> |
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4291 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. omnes, id est summa omnium |
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4292 <lb/>[<emph style="it">tr: |
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4293 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> is all, that is the sum of all. |
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4294 </emph>]<lb/> |
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4295 </s> |
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4296 </p> |
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4297 <p xml:lang="lat"> |
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4298 <s xml:id="echoid-s323" xml:space="preserve"> |
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|
4299 ita Vieta post δεδόμενα <lb/> |
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|
4300 in respons: pag. 29. |
|
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|
4301 <lb/>[<emph style="it">tr: |
|
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|
4302 thus Viete after δεδόμενα in |
|
22d6a63640c6
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|
4303 <emph style="it">Variorum Responsorum</emph> page 29. |
|
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|
4304 </emph>]<lb/> |
|
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|
4305 </s> |
|
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|
4306 </p> |
|
22d6a63640c6
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|
4307 <p xml:lang="lat"> |
|
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|
4308 <s xml:id="echoid-s324" xml:space="preserve"> |
|
22d6a63640c6
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|
4309 apud Vieta dicitur crementum. |
|
22d6a63640c6
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|
4310 <lb/>[<emph style="it">tr: |
|
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|
4311 in Viete this is said to be the increment. |
|
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|
4312 </emph>]<lb/> |
|
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|
4313 </s> |
|
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|
4314 </p> |
|
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|
4315 <pb file="add_6784_f363v" o="363v" n="726"/> |
|
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|
4316 <pb file="add_6784_f364" o="364" n="727"/> |
|
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|
4317 <div xml:id="echoid-div61" type="page_commentary" level="2" n="61"> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4318 <p> |
|
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|
4319 <s xml:id="echoid-s325" xml:space="preserve">[<emph style="it">Note: |
|
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|
4320 <p> |
|
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|
4321 <s xml:id="echoid-s325" xml:space="preserve"> |
|
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|
4322 On this folio an expression that looks like <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mo>=</mo><mi>s</mi></mstyle></math> is to be read as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo lspace="0em" rspace="0em" maxsize="1">|</mo><mi>p</mi><mo>-</mo><mi>s</mi><mo lspace="0em" rspace="0em" maxsize="1">|</mo></mstyle></math>. |
|
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|
4323 </s> |
|
22d6a63640c6
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|
4324 </p> |
|
22d6a63640c6
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diff
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|
4325 </emph>] |
|
22d6a63640c6
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|
4326 <lb/><lb/></s></p></div> |
|
22d6a63640c6
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|
4327 <head xml:id="echoid-head108" xml:space="preserve" xml:lang="lat"> |
|
22d6a63640c6
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diff
changeset
|
4328 De progressionibus. <lb/> |
|
22d6a63640c6
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casties
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diff
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|
4329 finitis & infinitis. |
|
22d6a63640c6
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|
4330 <lb/>[<emph style="it">tr: |
|
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diff
changeset
|
4331 On finite and infinite progressions |
|
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|
4332 </emph>]<lb/> |
|
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diff
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|
4333 </head> |
|
22d6a63640c6
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|
4334 <p xml:lang="lat"> |
|
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diff
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|
4335 <s xml:id="echoid-s327" xml:space="preserve"> |
|
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diff
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|
4336 linea infinite <emph style="super">longa</emph>quælibet = æqualis alicui, plano <lb/> |
|
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|
4337 solido. <lb/> |
|
22d6a63640c6
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diff
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|
4338 longo-solido. <lb/> |
|
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diff
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|
4339 plano-solido. <lb/> |
|
22d6a63640c6
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|
4340 solido-solido. &c. |
|
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|
4341 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
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|
4342 An infinite line of any length is equal to some plane, or solid, or solid-length, or solid-plane, or solid-solid, etc. |
|
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|
4343 </emph>]<lb/> |
|
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diff
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|
4344 </s> |
|
22d6a63640c6
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diff
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|
4345 </p> |
|
22d6a63640c6
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diff
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|
4346 <p xml:lang="lat"> |
|
22d6a63640c6
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diff
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|
4347 <s xml:id="echoid-s328" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
4348 linea infinite brevis quælibet = æqualis alicui, puncto. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
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|
4349 linea. <lb/> |
|
22d6a63640c6
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casties
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diff
changeset
|
4350 puncto-plano. <lb/> |
|
22d6a63640c6
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casties
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diff
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|
4351 puncto-solido. &c. <lb/> |
|
22d6a63640c6
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diff
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|
4352 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
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|
4353 Any infinitely short line is equal to some line-point, or plane-point, or solid-point, etc. |
|
22d6a63640c6
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diff
changeset
|
4354 </emph>]<lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
4355 </s> |
|
22d6a63640c6
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casties
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diff
changeset
|
4356 <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
4357 <s xml:id="echoid-s329" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4358 Quælibet punctum terminat progressionem. |
|
22d6a63640c6
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casties
parents:
diff
changeset
|
4359 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4360 Whatever point terminates the progression. |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4361 </emph>]<lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4362 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4363 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4364 <p xml:lang="lat"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4365 <s xml:id="echoid-s330" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4366 infinite numero puncta = lineæ <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4367 plano. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4368 solido. &c. |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4369 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
4370 an infinite number of points equal a line, or plane, or solid, etc. |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4371 </emph>]<lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4372 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4373 <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4374 <s xml:id="echoid-s331" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4375 linea signata <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4376 terminat <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4377 progressionem. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4378 ita planum signatum. |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4379 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4380 a designated line terminates the progression; similarly a designated plane, |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff
changeset
|
4381 </emph>]<lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4382 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4383 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4384 <p xml:lang="lat"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4385 <s xml:id="echoid-s332" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4386 hæc & alia huius generis <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
4387 consideranda. |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
4388 <lb/>[<emph style="it">tr: |
|
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4389 these and others of this kind may be considered. |
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4390 </emph>]<lb/> |
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4391 </s> |
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4392 </p> |
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4393 <pb file="add_6784_f364v" o="364v" n="728"/> |
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4394 <pb file="add_6784_f365" o="365" n="729"/> |
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4395 <pb file="add_6784_f365v" o="365v" n="730"/> |
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4396 <pb file="add_6784_f366" o="366" n="731"/> |
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4397 <pb file="add_6784_f366v" o="366v" n="732"/> |
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4398 <pb file="add_6784_f367" o="367" n="733"/> |
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4399 <pb file="add_6784_f367v" o="367v" n="734"/> |
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4400 <pb file="add_6784_f368" o="368" n="735"/> |
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|
4401 <pb file="add_6784_f368v" o="368v" n="736"/> |
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4402 <pb file="add_6784_f369" o="369" n="737"/> |
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4403 <div xml:id="echoid-div62" type="page_commentary" level="2" n="62"> |
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4404 <p> |
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4405 <s xml:id="echoid-s333" xml:space="preserve">[<emph style="it">Note: |
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|
4406 <p> |
|
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4407 <s xml:id="echoid-s333" xml:space="preserve"> |
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|
4408 This page contains a symbolic version of Euclid Book II, Proposition 11: <lb/> |
|
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|
4409 II.11. To cut a given straight line so that the rectangle contained by the whole |
|
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|
4410 and one of the segments equals the square on the remaining segment. |
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4411 </s> |
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|
4412 </p> |
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|
4413 </emph>] |
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4414 <lb/><lb/></s></p></div> |
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|
4415 <head xml:id="echoid-head109" xml:space="preserve" xml:lang="lat"> |
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|
4416 propositiones 2<emph style="super">i</emph> Euclidis |
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|
4417 <lb/>[<emph style="it">tr: |
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|
4418 Propositions from the second book of Euclid |
|
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4419 </emph>]<lb/> |
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4420 </head> |
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4421 <pb file="add_6784_f369v" o="369v" n="738"/> |
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|
4422 <pb file="add_6784_f370" o="370" n="739"/> |
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|
4423 <pb file="add_6784_f370v" o="370v" n="740"/> |
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|
4424 <pb file="add_6784_f371" o="371" n="741"/> |
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|
4425 <pb file="add_6784_f371v" o="371v" n="742"/> |
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|
4426 <pb file="add_6784_f372" o="372" n="743"/> |
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|
4427 <pb file="add_6784_f372v" o="372v" n="744"/> |
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|
4428 <pb file="add_6784_f373" o="373" n="745"/> |
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|
4429 <pb file="add_6784_f373v" o="373v" n="746"/> |
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|
4430 <pb file="add_6784_f374" o="374" n="747"/> |
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|
4431 <pb file="add_6784_f374v" o="374v" n="748"/> |
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|
4432 <pb file="add_6784_f375" o="375" n="749"/> |
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|
4433 <pb file="add_6784_f375v" o="375v" n="750"/> |
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|
4434 <pb file="add_6784_f376" o="376" n="751"/> |
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|
4435 <pb file="add_6784_f376v" o="376v" n="752"/> |
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|
4436 <pb file="add_6784_f377" o="377" n="753"/> |
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|
4437 <pb file="add_6784_f377v" o="377v" n="754"/> |
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|
4438 <pb file="add_6784_f378" o="378" n="755"/> |
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|
4439 <pb file="add_6784_f378v" o="378v" n="756"/> |
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|
4440 <pb file="add_6784_f379" o="379" n="757"/> |
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|
4441 <pb file="add_6784_f379v" o="379v" n="758"/> |
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|
4442 <pb file="add_6784_f380" o="380" n="759"/> |
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|
4443 <pb file="add_6784_f380v" o="380v" n="760"/> |
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|
4444 <pb file="add_6784_f381" o="381" n="761"/> |
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|
4445 <pb file="add_6784_f381v" o="381v" n="762"/> |
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|
4446 <pb file="add_6784_f382" o="382" n="763"/> |
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|
4447 <pb file="add_6784_f382v" o="382v" n="764"/> |
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|
4448 <pb file="add_6784_f383" o="383" n="765"/> |
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|
4449 <pb file="add_6784_f383v" o="383v" n="766"/> |
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|
4450 <pb file="add_6784_f384" o="384" n="767"/> |
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|
4451 <pb file="add_6784_f384v" o="384v" n="768"/> |
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|
4452 <pb file="add_6784_f385" o="385" n="769"/> |
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|
4453 <pb file="add_6784_f385v" o="385v" n="770"/> |
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|
4454 <pb file="add_6784_f386" o="386" n="771"/> |
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|
4455 <pb file="add_6784_f386v" o="386v" n="772"/> |
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|
4456 <pb file="add_6784_f387" o="387" n="773"/> |
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4457 <pb file="add_6784_f387v" o="387v" n="774"/> |
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|
4458 <pb file="add_6784_f388" o="388" n="775"/> |
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4459 <pb file="add_6784_f388v" o="388v" n="776"/> |
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|
4460 <pb file="add_6784_f389" o="389" n="777"/> |
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|
4461 <pb file="add_6784_f389v" o="389v" n="778"/> |
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|
4462 <pb file="add_6784_f390" o="390" n="779"/> |
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4463 <pb file="add_6784_f390v" o="390v" n="780"/> |
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|
4464 <pb file="add_6784_f391" o="391" n="781"/> |
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|
4465 <pb file="add_6784_f391v" o="391v" n="782"/> |
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|
4466 <pb file="add_6784_f392" o="392" n="783"/> |
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|
4467 <pb file="add_6784_f392v" o="392v" n="784"/> |
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|
4468 <pb file="add_6784_f393" o="393" n="785"/> |
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|
4469 <pb file="add_6784_f393v" o="393v" n="786"/> |
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|
4470 <pb file="add_6784_f394" o="394" n="787"/> |
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4471 <pb file="add_6784_f394v" o="394v" n="788"/> |
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4472 <pb file="add_6784_f395" o="395" n="789"/> |
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4473 <pb file="add_6784_f395v" o="395v" n="790"/> |
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4474 <pb file="add_6784_f396" o="396" n="791"/> |
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4475 <pb file="add_6784_f396v" o="396v" n="792"/> |
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4476 <pb file="add_6784_f397" o="397" n="793"/> |
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4477 <pb file="add_6784_f397v" o="397v" n="794"/> |
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4478 <head xml:id="echoid-head110" xml:space="preserve" xml:lang="lat"> |
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4479 1.) De reductione æquationum |
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4480 <lb/>[<emph style="it">tr: |
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4481 On the reduction of equations |
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4482 </emph>]<lb/> |
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4483 </head> |
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4484 <pb file="add_6784_f398" o="398" n="795"/> |
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4485 <head xml:id="echoid-head111" xml:space="preserve"> |
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4486 3.) |
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4487 </head> |
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4488 <pb file="add_6784_f398v" o="398v" n="796"/> |
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4489 <pb file="add_6784_f399" o="399" n="797"/> |
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4490 <pb file="add_6784_f399v" o="399v" n="798"/> |
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4491 <pb file="add_6784_f400" o="400" n="799"/> |
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4492 <pb file="add_6784_f400v" o="400v" n="800"/> |
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4493 <head xml:id="echoid-head112" xml:space="preserve" xml:lang="lat"> |
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4494 1)B) De reductione æquationum |
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4495 <lb/>[<emph style="it">tr: |
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4496 On the reduction of equations |
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4497 </emph>]<lb/> |
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4498 </head> |
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4499 <pb file="add_6784_f401" o="401" n="801"/> |
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4500 <pb file="add_6784_f401v" o="401v" n="802"/> |
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4501 <div xml:id="echoid-div63" type="page_commentary" level="2" n="63"> |
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4502 <p> |
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4503 <s xml:id="echoid-s335" xml:space="preserve">[<emph style="it">Note: |
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4504 <p> |
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4505 <s xml:id="echoid-s335" xml:space="preserve"> |
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4506 Here Harriot solves the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>5</mn><mo>=</mo><mn>6</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math> (in modern notation, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>5</mn><mo>=</mo><mn>6</mn><mi>x</mi><mo>-</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle></math>) |
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4507 for the roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>+</mo><msqrt><mrow><mo>-</mo><mn>1</mn></mrow></msqrt></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>-</mo><msqrt><mrow><mo>-</mo><mn>1</mn></mrow></msqrt></mstyle></math>. He then checks by multiplication |
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4508 that these valus do indeed satisfy the equation. |
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4509 </s> |
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4510 </p> |
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4511 </emph>] |
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4512 <lb/><lb/></s></p></div> |
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4513 <pb file="add_6784_f402" o="402" n="803"/> |
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4514 <div xml:id="echoid-div64" type="page_commentary" level="2" n="64"> |
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4515 <p> |
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4516 <s xml:id="echoid-s337" xml:space="preserve">[<emph style="it">Note: |
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4517 <p> |
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4518 <s xml:id="echoid-s337" xml:space="preserve"> |
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4519 Powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>2</mn><mn>0</mn><mo>+</mo><mn>4</mn><mo maxsize="1">)</mo></mstyle></math> up to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>2</mn><mn>0</mn><mo>+</mo><mn>4</mn><mrow><msup><mo maxsize="1">)</mo><mn>5</mn></msup></mrow></mstyle></math> following the pattern laid out in Add MS 6782, f. 276. <lb/> |
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4520 A calculation below each box gives the sum of the figures contained in it. |
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4521 </s> |
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4522 </p> |
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4523 </emph>] |
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4524 <lb/><lb/></s></p></div> |
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4525 <pb file="add_6784_f402v" o="402v" n="804"/> |
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4526 <div xml:id="echoid-div65" type="page_commentary" level="2" n="65"> |
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4527 <p> |
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4528 <s xml:id="echoid-s339" xml:space="preserve">[<emph style="it">Note: |
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4529 <p> |
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4530 <s xml:id="echoid-s339" xml:space="preserve"> |
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4531 The calculations from the previous page (Add MS 6784, f. 402) are checked by root extractions |
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4532 </s> |
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4533 </p> |
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4534 </emph>] |
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4535 <lb/><lb/></s></p></div> |
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4536 <p> |
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4537 <s xml:id="echoid-s341" xml:space="preserve"> |
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4538 The extraction <lb/> |
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4539 of the roots. |
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4540 </s> |
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4541 </p> |
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4542 <pb file="add_6784_f403" o="403" n="805"/> |
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4543 <pb file="add_6784_f403v" o="403v" n="806"/> |
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4544 <pb file="add_6784_f404" o="404" n="807"/> |
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4545 <div xml:id="echoid-div66" type="page_commentary" level="2" n="66"> |
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4546 <p> |
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4547 <s xml:id="echoid-s342" xml:space="preserve">[<emph style="it">Note: |
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4548 <p> |
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4549 <s xml:id="echoid-s342" xml:space="preserve"> |
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4550 Third, fourth, and fifth powers of (20 + 4). <lb/> |
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4551 The binomial coefficients 3, 3 and 4, 6, 4 and 5, 10, 10, 5, |
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|
4552 appear amongst the numbers in the rightmost column. |
|
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|
4553 </s> |
|
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|
4554 </p> |
|
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|
4555 </emph>] |
|
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|
4556 <lb/><lb/></s></p></div> |
|
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|
4557 <pb file="add_6784_f404v" o="404v" n="808"/> |
|
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|
4558 <p> |
|
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|
4559 <s xml:id="echoid-s344" xml:space="preserve"> |
|
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|
4560 The doctrine of Algebraycall nombers is but <lb/> |
|
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|
4561 the doctrined of such continuall proportionalles of <lb/> |
|
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|
4562 which a unite is the first. |
|
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|
4563 </s> |
|
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|
4564 </p> |
|
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|
4565 <p> |
|
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|
4566 <s xml:id="echoid-s345" xml:space="preserve"> |
|
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|
4567 A unite being the first of continuall proportionalles; the second is <lb/> |
|
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|
4568 called a roote: because the third wilbe always a square: & the fourth <lb/> |
|
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|
4569 <emph style="st">third</emph> a cube, as Euclide demonstrateth. |
|
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|
4570 </s> |
|
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|
4571 <s xml:id="echoid-s346" xml:space="preserve"> |
|
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|
4572 The names of the other proportionalles <lb/> |
|
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|
4573 following are all compounded of squares, or cubes or both according <lb/> |
|
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|
4574 to Diophantus & others which follow him. |
|
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|
4575 </s> |
|
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|
4576 <s xml:id="echoid-s347" xml:space="preserve"> |
|
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|
4577 Some or other of the most parte of the later <lb/> |
|
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|
4578 writers gave the name of surdsolidus, of which the first or simple sursolid <lb/> |
|
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|
4579 is the sixt proportionall. &c. |
|
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|
4580 </s> |
|
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|
4581 </p> |
|
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|
4582 <p> |
|
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|
4583 <s xml:id="echoid-s348" xml:space="preserve"> |
|
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|
4584 Any nomber may be <emph style="super">any</emph> terme proportinall in a continuall progression <lb/> |
|
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|
4585 from a unite. |
|
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|
4586 </s> |
|
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|
4587 <s xml:id="echoid-s349" xml:space="preserve"> |
|
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|
4588 If the nomber terme be the second, the third is gotten by <lb/> |
|
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|
4589 multiplying the nomber into him self. |
|
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|
4590 </s> |
|
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|
4591 <s xml:id="echoid-s350" xml:space="preserve"> |
|
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|
4592 & the fourth by multiplying the <lb/> |
|
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|
4593 third by the second & so forth. |
|
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|
4594 </s> |
|
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|
4595 <s xml:id="echoid-s351" xml:space="preserve"> |
|
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|
4596 as also <emph style="super">by</emph> the doctrine of progression <lb/> |
|
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|
4597 any terme that is found another may be gotten compendiously <lb/> |
|
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|
4598 without continuall multiplications. |
|
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|
4599 </s> |
|
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|
4600 </p> |
|
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|
4601 <p> |
|
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|
4602 <s xml:id="echoid-s352" xml:space="preserve"> |
|
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|
4603 If a nomber that is known & designed to be the third, fourth, <lb/> |
|
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|
4604 or fifth or any other proportinall of another denomination: the <lb/> |
|
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|
4605 doctrine to find the second is that which is called the extraction <lb/> |
|
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|
4606 of the roote, which is taught in these papers. |
|
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|
4607 </s> |
|
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|
4608 </p> |
|
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|
4609 <p> |
|
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|
4610 <s xml:id="echoid-s353" xml:space="preserve"> |
|
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|
4611 The second proportionall is also called the first dignity, & the third the <lb/> |
|
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|
4612 second dignity, & the fourth the third dignity &c. |
|
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|
4613 </s> |
|
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|
4614 </p> |
|
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|
4615 <p> |
|
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|
4616 <s xml:id="echoid-s354" xml:space="preserve"> |
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|
4617 The third is also called the first power; the 4th the second power &c. |
|
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|
4618 </s> |
|
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|
4619 </p> |
|
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|
4620 <p> |
|
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|
4621 <s xml:id="echoid-s355" xml:space="preserve"> |
|
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|
4622 The first proportionall <lb/> |
|
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|
4623 is a unite. |
|
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|
4624 </s> |
|
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|
4625 </p> |
|
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|
4626 <p> |
|
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|
4627 <s xml:id="echoid-s356" xml:space="preserve"> |
|
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|
4628 The first dignity is <lb/> |
|
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|
4629 the second proportionall, <lb/> |
|
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|
4630 called a roote. |
|
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|
4631 </s> |
|
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|
4632 </p> |
|
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|
4633 <p> |
|
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|
4634 <s xml:id="echoid-s357" xml:space="preserve"> |
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|
4635 The first power is the <lb/> |
|
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|
4636 third proportionall <lb/> |
|
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|
4637 <emph style="st">called a square</emph> <lb/> |
|
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|
4638 or second Dignity <lb/> |
|
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|
4639 called a square. |
|
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|
4640 </s> |
|
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|
4641 </p> |
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4642 <p> |
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4643 <s xml:id="echoid-s358" xml:space="preserve"> |
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4644 The first solid is the <lb/> |
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4645 fourth proprtionall: <lb/> |
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4646 The third dignity: & <lb/> |
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4647 The second power, <lb/> |
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4648 called a cube. |
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4649 </s> |
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4650 </p> |
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4651 <p> |
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4652 <s xml:id="echoid-s359" xml:space="preserve"> |
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4653 The pythagoreans <lb/> |
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4654 did call 4 the first solid <lb/> |
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4655 as Boethius relateth. |
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4656 </s> |
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4657 <lb/> |
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4658 <s xml:id="echoid-s360" xml:space="preserve"> |
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4659 The nomber serveth to be, because pyramides are prime solids <lb/> |
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4660 & 4 amongst nombers is the first pyramide. |
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4661 </s> |
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4662 </p> |
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4663 <pb file="add_6784_f405" o="405" n="809"/> |
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4664 <pb file="add_6784_f405v" o="405v" n="810"/> |
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4665 <pb file="add_6784_f406" o="406" n="811"/> |
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4666 <pb file="add_6784_f406v" o="406v" n="812"/> |
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4667 <pb file="add_6784_f407" o="407" n="813"/> |
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4668 <div xml:id="echoid-div67" type="page_commentary" level="2" n="67"> |
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4669 <p> |
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4670 <s xml:id="echoid-s361" xml:space="preserve">[<emph style="it">Note: |
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4671 <p> |
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4672 <s xml:id="echoid-s361" xml:space="preserve"> |
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4673 Here Harriot demonstrates that multiplication by 9 increases the number of digits by one |
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4674 as far as the 21st power but not at the 22nd power. |
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4675 Thus the number of digits alone is no guide to the size of the root. |
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4676 </s> |
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4677 </p> |
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4678 </emph>] |
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4679 <lb/><lb/></s></p></div> |
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4680 <p> |
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4681 <s xml:id="echoid-s363" xml:space="preserve"> |
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4682 An induction to prove that <lb/> |
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4683 to pricke the second figure for <lb/> |
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4684 the extraction of square rootes <lb/> |
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4685 & the third for cubes & 4th <lb/> |
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4686 for biquadrates etc. according <lb/> |
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4687 to the nomber of figures that <lb/> |
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4688 the greatest figure 9 doth <lb/> |
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4689 produce is no rule. |
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4690 </s> |
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4691 <s xml:id="echoid-s364" xml:space="preserve"> |
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|
4692 for we <lb/> |
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|
4693 may see how it breaketh in <lb/> |
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4694 the 22th <emph style="st">proportionall</emph> dignity & so <lb/> |
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|
4695 forwarde. |
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|
4696 </s> |
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|
4697 <s xml:id="echoid-s365" xml:space="preserve"> |
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|
4698 but the true case <lb/> |
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|
4699 of such pricking appeareth <lb/> |
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|
4700 out <emph style="super">of</emph> the speciosa genesis which <lb/> |
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|
4701 is in an other paper arranged. |
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|
4702 </s> |
|
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|
4703 </p> |
|
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|
4704 <pb file="add_6784_f407v" o="407v" n="814"/> |
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|
4705 <pb file="add_6784_f408" o="408" n="815"/> |
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|
4706 <div xml:id="echoid-div68" type="page_commentary" level="2" n="68"> |
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|
4707 <p> |
|
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|
4708 <s xml:id="echoid-s366" xml:space="preserve">[<emph style="it">Note: |
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|
4709 <p> |
|
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|
4710 <s xml:id="echoid-s366" xml:space="preserve"> |
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|
4711 Calculation of powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>a</mi></mstyle></math> to show how the digits of a three-digit number are distributed in the sum. |
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|
4712 </s> |
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|
4713 </p> |
|
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|
4714 </emph>] |
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|
4715 <lb/><lb/></s></p></div> |
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|
4716 <p> |
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|
4717 <s xml:id="echoid-s368" xml:space="preserve"> |
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|
4718 If the roote to be extracted be three figures <lb/> |
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|
4719 the two first as one may here see are to be had <lb/> |
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|
4720 according to the generall rule, the next is <lb/> |
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|
4721 also to be gotten really after the same manner <lb/> |
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|
4722 that <emph style="super">is</emph> supposing the two first to be as one, & that <lb/> |
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|
4723 which foloweth, the second; although in appearance <lb/> |
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|
4724 & expressing by wordes it seems otherwise. |
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|
4725 </s> |
|
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|
4726 </p> |
|
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|
4727 <pb file="add_6784_f408v" o="408v" n="816"/> |
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|
4728 <pb file="add_6784_f409" o="409" n="817"/> |
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4729 <div xml:id="echoid-div69" type="page_commentary" level="2" n="69"> |
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4730 <p> |
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4731 <s xml:id="echoid-s369" xml:space="preserve">[<emph style="it">Note: |
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4732 <p> |
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4733 <s xml:id="echoid-s369" xml:space="preserve"> |
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4734 Powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>–</mo><mi>c</mi><mo maxsize="1">)</mo></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mo maxsize="1">)</mo></mstyle></math>. |
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4735 </s> |
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4736 </p> |
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4737 </emph>] |
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4738 <lb/><lb/></s></p></div> |
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4739 <pb file="add_6784_f409v" o="409v" n="818"/> |
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4740 <pb file="add_6784_f410" o="410" n="819"/> |
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4741 <pb file="add_6784_f410v" o="410v" n="820"/> |
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4742 <pb file="add_6784_f411" o="411" n="821"/> |
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4743 <div xml:id="echoid-div70" type="page_commentary" level="2" n="70"> |
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4744 <p> |
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4745 <s xml:id="echoid-s371" xml:space="preserve">[<emph style="it">Note: |
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4746 <p> |
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4747 <s xml:id="echoid-s371" xml:space="preserve"> |
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4748 Here and on folio Add MS 6784, f. 412, Harriot shows that the product of two or three unequal parts |
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4749 is always less than the product of the same number of equal parts. |
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4750 </s> |
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4751 </p> |
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4752 </emph>] |
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4753 <lb/><lb/></s></p></div> |
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4754 <head xml:id="echoid-head113" xml:space="preserve" xml:lang="lat"> |
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4755 1<emph style="super">o</emph>. de bisectione. |
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4756 <lb/>[<emph style="it">tr: |
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4757 1. on bisection |
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4758 </emph>]<lb/> |
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4759 </head> |
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4760 <p xml:lang="lat"> |
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4761 <s xml:id="echoid-s373" xml:space="preserve"> |
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4762 Sit: tota linea. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>b</mi></mstyle></math>. <lb/> |
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4763 vel duæ æquales partes. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>b</mi></mstyle></math>.<lb/> |
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4764 magnitudo facta ab illis <lb/> |
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4765 erit quadratum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math>. |
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4766 <lb/>[<emph style="it">tr: |
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4767 Let the total line be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>b</mi></mstyle></math> <lb/> |
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4768 or two equal parts <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>b</mi></mstyle></math>, <lb/> |
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4769 the size of their product will be the square <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math>. |
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4770 </emph>]<lb/> |
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4771 </s> |
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4772 <lb/> |
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4773 <s xml:id="echoid-s374" xml:space="preserve"> |
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4774 Sint inæquales partes. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> <lb/> |
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4775 et: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>c</mi></mstyle></math> |
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4776 <lb/>[<emph style="it">tr: |
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4777 Let there be unequal parts <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>c</mi></mstyle></math>. |
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4778 </emph>]<lb/> |
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|
4779 </s> |
|
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4780 <lb/> |
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4781 <s xml:id="echoid-s375" xml:space="preserve"> |
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|
4782 magnitudo facta: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>c</mi><mo><</mo><mi>b</mi><mi>b</mi></mstyle></math>. |
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4783 <lb/>[<emph style="it">tr: |
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|
4784 the size of the product is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>c</mi><mo><</mo><mi>b</mi><mi>b</mi></mstyle></math>. |
|
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|
4785 </emph>]<lb/> |
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|
4786 </s> |
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|
4787 </p> |
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4788 <p xml:lang="lat"> |
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4789 <s xml:id="echoid-s376" xml:space="preserve"> |
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|
4790 Si linea dividatur utcunque in tot <lb/> |
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4791 partes inæquales, quot æquales: <lb/> |
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|
4792 Magnitudo facta ab inæquali-<lb/> |
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|
4793 bus, minor est illa quæ facta <lb/> |
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4794 ab æqualibus. |
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|
4795 <lb/>[<emph style="it">tr: |
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|
4796 If a line is divided in any way into as many unequal parts as equal parts, |
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|
4797 the size of the product of the unequal parts is less than the product of the equal parts. |
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|
4798 </emph>]<lb/> |
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4799 </s> |
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|
4800 </p> |
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|
4801 <p xml:lang="lat"> |
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|
4802 <s xml:id="echoid-s377" xml:space="preserve"> |
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|
4803 vel: |
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|
4804 <lb/>[<emph style="it">tr: |
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|
4805 or: |
|
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|
4806 </emph>]<lb/> |
|
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|
4807 </s> |
|
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4808 <lb/> |
|
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|
4809 <s xml:id="echoid-s378" xml:space="preserve"> |
|
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|
4810 Si aggregatum linearum inæqualium æqueretur <lb/> |
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|
4811 aggregato tot æqualium: Magnitudo facta &c. |
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4812 <lb/>[<emph style="it">tr: |
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|
4813 If the sum of the unnequal lines is equal to the sum of as many equals, the size of the product etc. |
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|
4814 </emph>]<lb/> |
|
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|
4815 </s> |
|
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|
4816 </p> |
|
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|
4817 <p xml:lang="lat"> |
|
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|
4818 <s xml:id="echoid-s379" xml:space="preserve"> |
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|
4819 etiam: |
|
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|
4820 <lb/>[<emph style="it">tr: |
|
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|
4821 also: |
|
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|
4822 </emph>]<lb/> |
|
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|
4823 </s> |
|
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|
4824 <lb/> |
|
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|
4825 <s xml:id="echoid-s380" xml:space="preserve"> |
|
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|
4826 plana facta ab inæqualibus <lb/> |
|
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|
4827 minora sunt quaduratis <lb/> |
|
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|
4828 facta ab æqualibus. |
|
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|
4829 <lb/>[<emph style="it">tr: |
|
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|
4830 planes made from unequals are less than squares made from equals. |
|
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|
4831 </emph>]<lb/> |
|
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|
4832 </s> |
|
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|
4833 </p> |
|
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|
4834 <head xml:id="echoid-head114" xml:space="preserve" xml:lang="lat"> |
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|
4835 2<emph style="it">o</emph>. De sectione in tres partes. |
|
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|
4836 <lb/>[<emph style="it">tr: |
|
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|
4837 2. On sectioning into three parts. |
|
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|
4838 </emph>]<lb/> |
|
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|
4839 </head> |
|
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|
4840 <p xml:lang="lat"> |
|
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|
4841 <s xml:id="echoid-s381" xml:space="preserve"> |
|
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|
4842 Casus primus |
|
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|
4843 <lb/>[<emph style="it">tr: |
|
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|
4844 First case. |
|
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|
4845 </emph>]<lb/> |
|
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|
4846 </s> |
|
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|
4847 <lb/> |
|
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|
4848 <s xml:id="echoid-s382" xml:space="preserve"> |
|
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|
4849 Sint tres inæquales partes. |
|
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|
4850 <lb/>[<emph style="it">tr: |
|
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|
4851 Let there be three unequalparts. |
|
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|
4852 </emph>]<lb/> |
|
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|
4853 </s> |
|
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|
4854 <lb/> |
|
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|
4855 <s xml:id="echoid-s383" xml:space="preserve"> |
|
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|
4856 magnitudo facta: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math> |
|
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|
4857 <lb/>[<emph style="it">tr: |
|
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|
4858 the size of the product is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math>. |
|
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|
4859 </emph>]<lb/> |
|
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|
4860 </s> |
|
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|
4861 </p> |
|
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|
4862 <p xml:lang="lat"> |
|
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|
4863 <s xml:id="echoid-s384" xml:space="preserve"> |
|
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|
4864 Tres æquales partes. |
|
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|
4865 <lb/>[<emph style="it">tr: |
|
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|
4866 Three equal parts. |
|
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|
4867 </emph>]<lb/> |
|
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|
4868 </s> |
|
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|
4869 <lb/> |
|
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|
4870 <s xml:id="echoid-s385" xml:space="preserve"> |
|
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diff
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|
4871 magnitudo facta <lb/> |
|
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|
4872 quæ cubus. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math> |
|
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|
4873 <lb/>[<emph style="it">tr: |
|
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|
4874 the size of the product which is a cube is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math>. |
|
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|
4875 </emph>]<lb/> |
|
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|
4876 </s> |
|
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|
4877 </p> |
|
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|
4878 <p xml:lang="lat"> |
|
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|
4879 <s xml:id="echoid-s386" xml:space="preserve"> |
|
22d6a63640c6
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|
4880 Casus 2<emph style="super">a</emph>. |
|
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|
4881 <lb/>[<emph style="it">tr: |
|
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|
4882 Case 2. |
|
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|
4883 </emph>]<lb/> |
|
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|
4884 </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4885 <lb/> |
|
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|
4886 <s xml:id="echoid-s387" xml:space="preserve"> |
|
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|
4887 Sint tres inæquales partes. |
|
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|
4888 <lb/>[<emph style="it">tr: |
|
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|
4889 Let there be three unequal parts. |
|
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|
4890 </emph>]<lb/> |
|
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|
4891 </s> |
|
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|
4892 <lb/> |
|
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|
4893 <s xml:id="echoid-s388" xml:space="preserve"> |
|
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|
4894 magnitudo facta. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>. |
|
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|
4895 <lb/>[<emph style="it">tr: |
|
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|
4896 the size of the product is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>. |
|
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4897 </emph>]<lb/> |
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4898 </s> |
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4899 </p> |
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4900 <p xml:lang="lat"> |
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4901 <s xml:id="echoid-s389" xml:space="preserve"> |
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|
4902 Tres æquales partes. |
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4903 <lb/>[<emph style="it">tr: |
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4904 Three equal parts. |
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4905 </emph>]<lb/> |
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4906 </s> |
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4907 <lb/> |
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4908 <s xml:id="echoid-s390" xml:space="preserve"> |
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4909 magnitudo facta <lb/> |
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4910 quæ cubus. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>. |
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4911 <lb/>[<emph style="it">tr: |
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4912 the size of the product which is a cube is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>. |
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4913 </emph>]<lb/> |
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4914 </s> |
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4915 </p> |
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4916 <pb file="add_6784_f411v" o="411v" n="822"/> |
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4917 <div xml:id="echoid-div71" type="page_commentary" level="2" n="71"> |
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4918 <p> |
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4919 <s xml:id="echoid-s391" xml:space="preserve">[<emph style="it">Note: |
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|
4920 <p> |
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4921 <s xml:id="echoid-s391" xml:space="preserve"> |
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|
4922 Note the combinations of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> (greater than), <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi></mstyle></math> (less than), and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> (equals), |
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4923 and of the symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo><</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>></mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo></mstyle></math> in the lower part of the page. |
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4924 </s> |
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|
4925 </p> |
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|
4926 </emph>] |
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4927 <lb/><lb/></s></p></div> |
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4928 <pb file="add_6784_f412" o="412" n="823"/> |
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4929 <div xml:id="echoid-div72" type="page_commentary" level="2" n="72"> |
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4930 <p> |
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4931 <s xml:id="echoid-s393" xml:space="preserve">[<emph style="it">Note: |
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4932 <p> |
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4933 <s xml:id="echoid-s393" xml:space="preserve"> |
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4934 The continuation of Add MS 6784, f. 411. |
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4935 </s> |
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4936 </p> |
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|
4937 </emph>] |
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4938 <lb/><lb/></s></p></div> |
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4939 <p xml:lang="lat"> |
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4940 <s xml:id="echoid-s395" xml:space="preserve"> |
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|
4941 Casus 3<emph style="super">a</emph>. |
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4942 <lb/>[<emph style="it">tr: |
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|
4943 Case 3. |
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4944 </emph>]<lb/> |
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4945 </s> |
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4946 <lb/> |
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4947 <s xml:id="echoid-s396" xml:space="preserve"> |
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4948 Sint tres inæquales partes. |
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4949 <lb/>[<emph style="it">tr: |
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|
4950 Let there be three unequal parts. |
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|
4951 </emph>]<lb/> |
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4952 </s> |
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4953 <lb/> |
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4954 <s xml:id="echoid-s397" xml:space="preserve"> |
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|
4955 magnitudo facta. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>. |
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|
4956 <lb/>[<emph style="it">tr: |
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|
4957 the size of the product is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>. |
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|
4958 </emph>]<lb/> |
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|
4959 </s> |
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|
4960 </p> |
|
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|
4961 <p xml:lang="lat"> |
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|
4962 <s xml:id="echoid-s398" xml:space="preserve"> |
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|
4963 Tres æquales partes. |
|
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|
4964 <lb/>[<emph style="it">tr: |
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|
4965 Three equal parts. |
|
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|
4966 </emph>]<lb/> |
|
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|
4967 </s> |
|
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|
4968 <lb/> |
|
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|
4969 <s xml:id="echoid-s399" xml:space="preserve"> |
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|
4970 magnitudo facta <lb/> |
|
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|
4971 quæ cubus. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>c</mi><mi>c</mi><mi>c</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>. |
|
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|
4972 <lb/>[<emph style="it">tr: |
|
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|
4973 the size of the product which is a cube is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>c</mi><mi>c</mi><mi>c</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>. |
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4974 </emph>]<lb/> |
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4975 </s> |
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4976 </p> |
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4977 <p xml:lang="lat"> |
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4978 <s xml:id="echoid-s400" xml:space="preserve"> |
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4979 Casus 4<emph style="super">a</emph>. |
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4980 <lb/>[<emph style="it">tr: |
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4981 Case 4. |
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4982 </emph>]<lb/> |
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4983 </s> |
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4984 <lb/> |
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4985 <s xml:id="echoid-s401" xml:space="preserve"> |
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4986 Sint tres inæquales partes. |
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4987 <lb/>[<emph style="it">tr: |
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4988 Let there be three unequal parts. |
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4989 </emph>]<lb/> |
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4990 </s> |
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4991 <lb/> |
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4992 <s xml:id="echoid-s402" xml:space="preserve"> |
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|
4993 magnitudo facta. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>d</mi><mo>-</mo><mn>9</mn><mo>,</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mn>9</mn><mo>,</mo><mi>b</mi><mi>d</mi><mi>c</mi></mstyle></math>. |
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4994 <lb/>[<emph style="it">tr: |
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4995 the size of the product is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>d</mi><mo>-</mo><mn>9</mn><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mn>9</mn><mi>b</mi><mi>d</mi><mi>c</mi></mstyle></math>. |
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4996 </emph>]<lb/> |
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4997 </s> |
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4998 </p> |
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4999 <p xml:lang="lat"> |
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5000 <s xml:id="echoid-s403" xml:space="preserve"> |
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|
5001 Tres æquales partes. |
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5002 <lb/>[<emph style="it">tr: |
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|
5003 Three equal parts. |
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5004 </emph>]<lb/> |
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|
5005 </s> |
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5006 <lb/> |
|
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5007 <s xml:id="echoid-s404" xml:space="preserve"> |
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|
5008 magnitudo facta <lb/> |
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|
5009 quæ cubus. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>d</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>d</mi><mi>d</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>d</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>d</mi><mo>-</mo><mn>9</mn><mo>,</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mn>9</mn><mo>,</mo><mi>b</mi><mi>d</mi><mi>c</mi></mstyle></math>. |
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5010 <lb/>[<emph style="it">tr: |
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|
5011 the size of the product which is a cube is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>d</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>d</mi><mi>d</mi><mo>+</mo><mi>d</mi><mi>d</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>d</mi><mo>-</mo><mn>9</mn><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mn>9</mn><mi>b</mi><mi>d</mi><mi>c</mi></mstyle></math>. |
|
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|
5012 </emph>]<lb/> |
|
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|
5013 </s> |
|
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|
5014 </p> |
|
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|
5015 <p xml:lang="lat"> |
|
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|
5016 <s xml:id="echoid-s405" xml:space="preserve"> |
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|
5017 Casus 5<emph style="super">a</emph>. <lb/> |
|
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|
5018 et ultimus. |
|
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|
5019 <lb/>[<emph style="it">tr: |
|
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|
5020 Case 5, and last. |
|
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|
5021 </emph>]<lb/> |
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|
5022 </s> |
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|
5023 <lb/> |
|
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|
5024 <s xml:id="echoid-s406" xml:space="preserve"> |
|
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|
5025 Sint tres inæquales partes. |
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|
5026 <lb/>[<emph style="it">tr: |
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|
5027 Let there be three unequal parts. |
|
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|
5028 </emph>]<lb/> |
|
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|
5029 </s> |
|
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|
5030 <lb/> |
|
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|
5031 <s xml:id="echoid-s407" xml:space="preserve"> |
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|
5032 magnitudo facta. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>d</mi><mo>-</mo><mn>9</mn><mo>,</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mn>9</mn><mo>,</mo><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>. |
|
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|
5033 <lb/>[<emph style="it">tr: |
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|
5034 the size of the product is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>d</mi><mo>-</mo><mn>9</mn><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mn>9</mn><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>. |
|
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|
5035 </emph>]<lb/> |
|
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|
5036 </s> |
|
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|
5037 </p> |
|
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|
5038 <p xml:lang="lat"> |
|
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|
5039 <s xml:id="echoid-s408" xml:space="preserve"> |
|
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|
5040 Tres æquales partes. |
|
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|
5041 <lb/>[<emph style="it">tr: |
|
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|
5042 Three equal parts. |
|
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|
5043 </emph>]<lb/> |
|
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|
5044 </s> |
|
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|
5045 <lb/> |
|
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|
5046 <s xml:id="echoid-s409" xml:space="preserve"> |
|
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|
5047 magnitudo facta <lb/> |
|
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|
5048 quæ cubus. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>d</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>d</mi><mi>d</mi><mo>-</mo><mi>d</mi><mi>d</mi><mi>d</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>d</mi><mo>-</mo><mn>9</mn><mo>,</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mn>9</mn><mo>,</mo><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>. |
|
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|
5049 <lb/>[<emph style="it">tr: |
|
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5050 the size of the product which is a cube is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>d</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>d</mi><mi>d</mi><mo>-</mo><mi>d</mi><mi>d</mi><mi>d</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>d</mi><mo>-</mo><mn>9</mn><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mn>9</mn><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>. |
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5051 </emph>]<lb/> |
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5052 </s> |
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5053 </p> |
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5054 <p xml:lang="lat"> |
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5055 <s xml:id="echoid-s410" xml:space="preserve"> |
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5056 nam: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>,</mo><mi>b</mi><mi>d</mi><mi>d</mi><mo>+</mo><mn>9</mn><mo>,</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>></mo><mi>d</mi><mi>d</mi><mi>d</mi><mo>+</mo><mn>9</mn><mi>b</mi><mi>c</mi><mi>d</mi><mo>.</mo></mstyle></math> |
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5057 <lb/>[<emph style="it">tr: |
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5058 for: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>b</mi><mi>d</mi><mi>d</mi><mo>+</mo><mn>9</mn><mi>b</mi><mi>c</mi><mi>c</mi><mo>></mo><mi>d</mi><mi>d</mi><mi>d</mi><mo>+</mo><mn>9</mn><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>. |
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5059 </emph>]<lb/> |
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5060 </s> |
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5061 </p> |
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5062 <pb file="add_6784_f412v" o="412v" n="824"/> |
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5063 <pb file="add_6784_f413" o="413" n="825"/> |
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5064 <pb file="add_6784_f413v" o="413v" n="826"/> |
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5065 <pb file="add_6784_f414" o="414" n="827"/> |
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5066 <div xml:id="echoid-div73" type="page_commentary" level="2" n="73"> |
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5067 <p> |
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5068 <s xml:id="echoid-s411" xml:space="preserve">[<emph style="it">Note: |
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5069 <p> |
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5070 <s xml:id="echoid-s411" xml:space="preserve"> |
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5071 Combinations of small numbers; see also Add MS 6784, f. 424. |
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5072 </s> |
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5073 </p> |
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5074 </emph>] |
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5075 <lb/><lb/></s></p></div> |
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5076 <pb file="add_6784_f414v" o="414v" n="828"/> |
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5077 <pb file="add_6784_f415" o="415" n="829"/> |
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5078 <div xml:id="echoid-div74" type="page_commentary" level="2" n="74"> |
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5079 <p> |
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5080 <s xml:id="echoid-s413" xml:space="preserve">[<emph style="it">Note: |
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5081 <p> |
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5082 <s xml:id="echoid-s413" xml:space="preserve"> |
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5083 This page summarizes in shorthand some rules that are written out in full in Harriot's treatise on cubic equations, |
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5084 on Add MS 6782, f. 186. <lb/> |
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5085 The abbreviations 'co:l' and 'co:pl' stand for 'longitudinal coefficient' and 'plane coefficient' respectively. |
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5086 In an equation of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>b</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>f</mi></mstyle></math>, the longitudinal coefficient is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> |
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5087 and the plane coefficient is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math>. |
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5088 Below the diagram Harriot has set out the different conditions under which such an equation can have three real roots, |
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5089 not necessarily distinct. The same sets of roots are also listed in Add MS 6783, f. 281. <lb/> |
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5090 The relevant equations are worked in full in sheets marked C, D, E, F, G |
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5091 (Add MS 6782, f. 315, f. 315v, f. 317, f. 318, f. 319), and also in Add MS 6783, f. 185. <lb/> |
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5092 </s> |
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5093 </p> |
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5094 </emph>] |
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5095 <lb/><lb/></s></p></div> |
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5096 <pb file="add_6784_f415v" o="415v" n="830"/> |
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5097 <pb file="add_6784_f416" o="416" n="831"/> |
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5098 <head xml:id="echoid-head115" xml:space="preserve" xml:lang="lat"> |
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5099 Ad generationes sequentium specierum æquationum |
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5100 <lb/>[<emph style="it">tr: |
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5101 On the generation of the following types of equation. |
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5102 </emph>]<lb/> |
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5103 </head> |
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5104 <p xml:lang="lat"> |
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5105 <s xml:id="echoid-s415" xml:space="preserve"> |
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5106 Æquatio <emph style="st">substantiva</emph> <lb/> |
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5107 parabolica. |
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5108 <lb/>[<emph style="it">tr: |
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5109 Parabolic equation |
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5110 </emph>]<lb/> |
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5111 </s> |
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5112 </p> |
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5113 <p xml:lang="lat"> |
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5114 <s xml:id="echoid-s416" xml:space="preserve"> |
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5115 Æquatio <emph style="st">adiectiva</emph> <emph style="super">hyperbolica</emph> <lb/> |
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5116 <emph style="st">sive additiva</emph>. |
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5117 Hyperbolic equation |
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5118 </s> |
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5119 </p> |
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5120 <p xml:lang="lat"> |
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5121 <s xml:id="echoid-s417" xml:space="preserve"> |
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5122 Æquatio <emph style="st">ablativa</emph> <emph style="super">elliptica</emph> <lb/> |
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5123 sive Bombellica. |
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5124 <lb/>[<emph style="it">tr: |
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5125 Elliptic, or Bombelli's, equation |
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5126 </emph>]<lb/> |
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5127 </s> |
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5128 </p> |
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5129 <p xml:lang="lat"> |
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5130 <s xml:id="echoid-s418" xml:space="preserve"> |
|
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|
5131 Ergo æquatio <emph style="st">nullitatis</emph> <emph style="st">prima</emph> <lb/> |
|
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|
5132 <emph style="st">sive [???]</emph> <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
5133 <emph style="st">sive</emph> primitiva. |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5134 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
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|
5135 Therefore the equation is primitive. |
|
22d6a63640c6
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|
5136 </emph>]<lb/> |
|
22d6a63640c6
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|
5137 </s> |
|
22d6a63640c6
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|
5138 <lb/> |
|
22d6a63640c6
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|
5139 <s xml:id="echoid-s419" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5140 Ergo verum quod proponebatur. |
|
22d6a63640c6
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|
5141 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
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|
5142 Therefore what was proposed is true. |
|
22d6a63640c6
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|
5143 </emph>]<lb/> |
|
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|
5144 </s> |
|
22d6a63640c6
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|
5145 </p> |
|
22d6a63640c6
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|
5146 <p xml:lang="lat"> |
|
22d6a63640c6
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|
5147 <s xml:id="echoid-s420" xml:space="preserve"> |
|
22d6a63640c6
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|
5148 Ad resolutiones sequentium specierum æquationum |
|
22d6a63640c6
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|
5149 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
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|
5150 On solving the following types of equation |
|
22d6a63640c6
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|
5151 </emph>]<lb/> |
|
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|
5152 </s> |
|
22d6a63640c6
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|
5153 <lb/> |
|
22d6a63640c6
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|
5154 <s xml:id="echoid-s421" xml:space="preserve"> |
|
22d6a63640c6
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|
5155 æquatio parabolica. |
|
22d6a63640c6
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|
5156 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
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|
5157 parabolic equation |
|
22d6a63640c6
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|
5158 </emph>]<lb/> |
|
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|
5159 </s> |
|
22d6a63640c6
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|
5160 <lb/> |
|
22d6a63640c6
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|
5161 <s xml:id="echoid-s422" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5162 æquatio hyperbolica. |
|
22d6a63640c6
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|
5163 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
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|
5164 hyperbolic equation |
|
22d6a63640c6
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|
5165 </emph>]<lb/> |
|
22d6a63640c6
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|
5166 </s> |
|
22d6a63640c6
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|
5167 <lb/> |
|
22d6a63640c6
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|
5168 <s xml:id="echoid-s423" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5169 æquatio elliptica. |
|
22d6a63640c6
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|
5170 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
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|
5171 elliptic equation |
|
22d6a63640c6
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|
5172 </emph>]<lb/> |
|
22d6a63640c6
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|
5173 </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
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|
5174 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
5175 <p xml:lang="lat"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5176 <s xml:id="echoid-s424" xml:space="preserve"> |
|
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
|
5177 rinus. |
|
22d6a63640c6
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|
5178 </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
|
5179 <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
5180 <s xml:id="echoid-s425" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
5181 prærinus. |
|
22d6a63640c6
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diff
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|
5182 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
5183 <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
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|
5184 <s xml:id="echoid-s426" xml:space="preserve"> |
|
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
|
5185 prinus. |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
5186 </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
|
5187 <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
5188 <s xml:id="echoid-s427" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff
changeset
|
5189 prino. |
|
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
|
5190 </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
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|
5191 <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
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|
5192 <s xml:id="echoid-s428" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5193 prinatus. prinatio. |
|
22d6a63640c6
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|
5194 </s> |
|
22d6a63640c6
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diff
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|
5195 <lb/> |
|
22d6a63640c6
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diff
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|
5196 <s xml:id="echoid-s429" xml:space="preserve"> |
|
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
|
5197 prinatimus. |
|
22d6a63640c6
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|
5198 </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
|
5199 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
5200 <pb file="add_6784_f416v" o="416v" n="832"/> |
|
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diff
changeset
|
5201 <pb file="add_6784_f417" o="417" n="833"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
5202 <pb file="add_6784_f417v" o="417v" n="834"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
5203 <pb file="add_6784_f418" o="418" n="835"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5204 <pb file="add_6784_f418v" o="418v" n="836"/> |
|
22d6a63640c6
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diff
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|
5205 <pb file="add_6784_f419" o="419" n="837"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5206 <pb file="add_6784_f419v" o="419v" n="838"/> |
|
22d6a63640c6
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|
5207 <pb file="add_6784_f420" o="420" n="839"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5208 <pb file="add_6784_f420v" o="420v" n="840"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5209 <pb file="add_6784_f421" o="421" n="841"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
5210 <pb file="add_6784_f421v" o="421v" n="842"/> |
|
22d6a63640c6
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|
5211 <pb file="add_6784_f422" o="422" n="843"/> |
|
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|
5212 <pb file="add_6784_f422v" o="422v" n="844"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5213 <pb file="add_6784_f423" o="423" n="845"/> |
|
22d6a63640c6
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changeset
|
5214 <div xml:id="echoid-div75" type="page_commentary" level="2" n="75"> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
5215 <p> |
|
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diff
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|
5216 <s xml:id="echoid-s430" xml:space="preserve">[<emph style="it">Note: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
5217 <p> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
5218 <s xml:id="echoid-s430" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5219 The polynomial <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>3</mn><mi>a</mi><mi>b</mi><mi>b</mi></mstyle></math> evaluated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn><mi>b</mi></mstyle></math>, ... , <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>7</mn><mi>b</mi></mstyle></math>. |
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5220 The resulting coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math> are listed in the table at the bottom of the page. |
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5221 Columns to the right list successive differences as far as the constant difference 6. |
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5222 The table has also been extrapolated upwards, giving rise to negative values in the first three columns. |
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5223 There is an error in the first column, however, which reading upwards should be: |
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5224 322, 110, 52, 18, 2, - 2, 0, 2, .... |
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5225 </s> |
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5226 </p> |
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5227 </emph>] |
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5228 <lb/><lb/></s></p></div> |
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5229 <pb file="add_6784_f423v" o="423v" n="846"/> |
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5230 <pb file="add_6784_f424" o="424" n="847"/> |
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5231 <div xml:id="echoid-div76" type="page_commentary" level="2" n="76"> |
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5232 <p> |
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5233 <s xml:id="echoid-s432" xml:space="preserve">[<emph style="it">Note: |
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5234 <p> |
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5235 <s xml:id="echoid-s432" xml:space="preserve"> |
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5236 Note various combinations of small numbers in the lower part of the page (see also Add MS 6784, f. 414). |
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5237 </s> |
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5238 </p> |
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5239 </emph>] |
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5240 <lb/><lb/></s></p></div> |
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5241 <pb file="add_6784_f424v" o="424v" n="848"/> |
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5242 <pb file="add_6784_f425" o="425" n="849"/> |
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5243 <pb file="add_6784_f425v" o="425v" n="850"/> |
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5244 <pb file="add_6784_f426" o="426" n="851"/> |
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5245 <pb file="add_6784_f426v" o="426v" n="852"/> |
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5246 <pb file="add_6784_f427" o="427" n="853"/> |
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5247 <pb file="add_6784_f427v" o="427v" n="854"/> |
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5248 <pb file="add_6784_f428" o="428" n="855"/> |
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5249 <div xml:id="echoid-div77" type="page_commentary" level="2" n="77"> |
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5250 <p> |
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5251 <s xml:id="echoid-s434" xml:space="preserve">[<emph style="it">Note: |
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5252 <p> |
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5253 <s xml:id="echoid-s434" xml:space="preserve"> |
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5254 Sums of some infinite geometric progressions. |
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5255 </s> |
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5256 </p> |
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5257 </emph>] |
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5258 <lb/><lb/></s></p></div> |
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5259 <pb file="add_6784_f428v" o="428v" n="856"/> |
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5260 <div xml:id="echoid-div78" type="page_commentary" level="2" n="78"> |
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5261 <p> |
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5262 <s xml:id="echoid-s436" xml:space="preserve">[<emph style="it">Note: |
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5263 <p> |
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5264 <s xml:id="echoid-s436" xml:space="preserve"> |
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5265 Triangles and circles filled with rectilinear figures (rectangles or triangles), |
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5266 in a way that can in principle be continued indefinitely. |
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5267 </s> |
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5268 </p> |
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5269 </emph>] |
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5270 <lb/><lb/></s></p></div> |
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5271 <pb file="add_6784_f429" o="429" n="857"/> |
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5272 <head xml:id="echoid-head116" xml:space="preserve" xml:lang="lat"> |
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5273 De infinitis. Ex ratione motus, temporis et spatij. |
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5274 <lb/>[<emph style="it">tr: |
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5275 On infinity. From the ratio of motion, time and space. |
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5276 </emph>]<lb/> |
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5277 </head> |
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5278 <p xml:lang="lat"> |
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5279 <s xml:id="echoid-s438" xml:space="preserve"> |
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5280 Vide <reg norm="Aristotle" type="abbr">Arist</reg>. lib. 6. tret. 23. <lb/> |
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5281 proclum de motu lib. 1. pro. 14. |
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5282 <lb/>[<emph style="it">tr: |
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5283 See Aristotle, Book 6, Treatise 23. <lb/> |
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5284 Proclus, <emph style="it">De motu</emph>, Book 1, Proposition 14. |
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5285 </emph>]<lb/> |
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5286 </s> |
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5287 </p> |
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5288 <p xml:lang="lat"> |
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5289 <s xml:id="echoid-s439" xml:space="preserve"> |
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5290 1. <lb/> |
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5291 Moveatur A corpus <lb/> |
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5292 per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> spatium in <lb/> |
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5293 tempore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math> atque sit <lb/> |
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5294 ille motus uniformis. |
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5295 <lb/>[<emph style="it">tr: |
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5296 Let a body A be moved through a distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> in a time <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math> and let that motion is uniform. |
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5297 </emph>]<lb/> |
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5298 </s> |
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5299 </p> |
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5300 <!-- text in first column --> |
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5301 <p xml:lang="lat"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5302 <s xml:id="echoid-s440" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5303 infinite <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5304 maximum |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5305 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5306 infinite maximum |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5307 </emph>]<lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5308 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5309 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5310 <p xml:lang="lat"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5311 <s xml:id="echoid-s441" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5312 minimum |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5313 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5314 minimum |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5315 </emph>]<lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5316 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5317 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5318 <p xml:lang="lat"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5319 <s xml:id="echoid-s442" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5320 indivisibile |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5321 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5322 an indivisible |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5323 </emph>]<lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5324 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5325 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5326 <p xml:lang="lat"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5327 <s xml:id="echoid-s443" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5328 punctum |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5329 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5330 a point |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5331 </emph>]<lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5332 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5333 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5334 <!-- text in second column --> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5335 <p xml:lang="lat"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5336 <s xml:id="echoid-s444" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5337 aliquod <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5338 infinite <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5339 maximum |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5340 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5341 infinite maximum |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5342 </emph>]<lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5343 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5344 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5345 <p xml:lang="lat"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5346 <s xml:id="echoid-s445" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5347 minimum <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5348 eadem <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5349 ratione |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5350 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5351 minimum in the same ratio |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5352 </emph>]<lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5353 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5354 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5355 <p xml:lang="lat"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5356 <s xml:id="echoid-s446" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5357 Indivisibile <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5358 eadem <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5359 ratione |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5360 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5361 An indivisble in the same ratio |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5362 </emph>]<lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5363 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5364 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5365 <p xml:lang="lat"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5366 <s xml:id="echoid-s447" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5367 Indivisibile <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5368 sed non punctum <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5369 vel instans ut alia <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5370 ratione inferetur. |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5371 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5372 And indivisble but not a point or an instant that can be inferred from the other ratio. |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5373 </emph>]<lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5374 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5375 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5376 <p xml:lang="lat"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5377 <s xml:id="echoid-s448" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5378 2. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5379 Moveatur A corpus per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> spatium <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5380 in tempore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi><mi>e</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> atque sit ille <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5381 motus uniformis. |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5382 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5383 Let a body A be moved thorugh a distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> in time <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi><mi>e</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and let that motion be uniform. |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5384 </emph>]<lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5385 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5386 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5387 <!-- text in first column --> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5388 <p xml:lang="lat"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5389 <s xml:id="echoid-s449" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5390 indivisibile |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
5391 <lb/>[<emph style="it">tr: |
|
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|
5392 an indivisible |
|
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|
5393 </emph>]<lb/> |
|
22d6a63640c6
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|
5394 </s> |
|
22d6a63640c6
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|
5395 </p> |
|
22d6a63640c6
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|
5396 <p xml:lang="lat"> |
|
22d6a63640c6
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|
5397 <s xml:id="echoid-s450" xml:space="preserve"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5398 punctum |
|
22d6a63640c6
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|
5399 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
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|
5400 a point |
|
22d6a63640c6
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|
5401 </emph>]<lb/> |
|
22d6a63640c6
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|
5402 </s> |
|
22d6a63640c6
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|
5403 </p> |
|
22d6a63640c6
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|
5404 <!-- text in second column --> |
|
22d6a63640c6
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|
5405 <p xml:lang="lat"> |
|
22d6a63640c6
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|
5406 <s xml:id="echoid-s451" xml:space="preserve"> |
|
22d6a63640c6
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|
5407 Indivisibile <lb/> |
|
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|
5408 eadem ratione |
|
22d6a63640c6
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|
5409 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
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|
5410 An indivisble in the same ratio |
|
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|
5411 </emph>]<lb/> |
|
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|
5412 </s> |
|
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|
5413 </p> |
|
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|
5414 <p xml:lang="lat"> |
|
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|
5415 <s xml:id="echoid-s452" xml:space="preserve"> |
|
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|
5416 Indivisibile quod <lb/> |
|
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|
5417 dimidium est <lb/> |
|
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|
5418 Indivisibilis ex <lb/> |
|
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|
5419 priori argumentatione. |
|
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|
5420 <lb/>[<emph style="it">tr: |
|
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|
5421 An indivisble whose half is indivisble by the previous argument. |
|
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|
5422 </emph>]<lb/> |
|
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|
5423 </s> |
|
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|
5424 </p> |
|
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|
5425 <p xml:lang="lat"> |
|
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|
5426 <s xml:id="echoid-s453" xml:space="preserve"> |
|
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|
5427 Ergo etiam: |
|
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|
5428 <lb/>[<emph style="it">tr: |
|
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|
5429 Therefore also |
|
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|
5430 </emph>]<lb/> |
|
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|
5431 </s> |
|
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|
5432 </p> |
|
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|
5433 <p xml:lang="lat"> |
|
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|
5434 <s xml:id="echoid-s454" xml:space="preserve"> |
|
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|
5435 Indivisibile quod <lb/> |
|
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|
5436 dimidium est <lb/> |
|
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|
5437 Indivisibilis ex <lb/> |
|
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|
5438 priori argumentatione. |
|
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|
5439 <lb/>[<emph style="it">tr: |
|
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|
5440 An indivisble whose half is indivisble by the previous argument. |
|
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|
5441 </emph>]<lb/> |
|
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|
5442 </s> |
|
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|
5443 </p> |
|
22d6a63640c6
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|
5444 <p xml:lang="lat"> |
|
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|
5445 <s xml:id="echoid-s455" xml:space="preserve"> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
5446 punctum |
|
22d6a63640c6
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|
5447 <lb/>[<emph style="it">tr: |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5448 a point |
|
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|
5449 </emph>]<lb/> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5450 </s> |
|
22d6a63640c6
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|
5451 </p> |
|
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|
5452 <p xml:lang="lat"> |
|
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|
5453 <s xml:id="echoid-s456" xml:space="preserve"> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5454 punctum |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5455 <lb/>[<emph style="it">tr: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5456 a point |
|
22d6a63640c6
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|
5457 </emph>]<lb/> |
|
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|
5458 </s> |
|
22d6a63640c6
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|
5459 </p> |
|
22d6a63640c6
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|
5460 <p xml:lang="lat"> |
|
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|
5461 <s xml:id="echoid-s457" xml:space="preserve"> |
|
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diff
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|
5462 Ergo punctum quod ponebatur esse <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
5463 indivisbile, alia ratione inferetur <lb/> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
5464 Divisibile, et sic in infinitum. |
|
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|
5465 <lb/>[<emph style="it">tr: |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5466 Therefore a point that can be supposed indivisble, is inferred from the other ratio to be divisible, |
|
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|
5467 and thus infinitely. |
|
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|
5468 </emph>]<lb/> |
|
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|
5469 </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5470 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
5471 <pb file="add_6784_f429v" o="429v" n="858"/> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5472 <pb file="add_6784_f430" o="430" n="859"/> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5473 <div xml:id="echoid-div79" type="page_commentary" level="2" n="79"> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5474 <p> |
|
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diff
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|
5475 <s xml:id="echoid-s458" xml:space="preserve">[<emph style="it">Note: |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5476 <p> |
|
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|
5477 <s xml:id="echoid-s458" xml:space="preserve"> |
|
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|
5478 Triangles transformed to spirals. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
5479 See also Add MS 6785, f. 437 and Add MS 6784, f. 246, f. 247, f. 248. |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5480 </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5481 </p> |
|
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|
5482 </emph>] |
|
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5483 <lb/><lb/></s></p></div> |
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5484 <pb file="add_6784_f430v" o="430v" n="860"/> |
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5485 <pb file="add_6784_f431" o="431" n="861"/> |
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5486 <pb file="add_6784_f431v" o="431v" n="862"/> |
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5487 </div> |
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5488 </text> |
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5489 </echo> |