Mercurial > hg > mpdl-xml-content
annotate texts/XML/echo/en/Harriot_Add_MS_6787_MAH52R5E.xml @ 29:90b1eda1b0a9
Some new special instructions
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
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| date | Fri, 02 Dec 2016 14:37:22 +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:MAH52R5E.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. 6787</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/MAH52R5E</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_6787_f001" o="1" n="1"/> |
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18 <pb file="add_6787_f001v" o="1v" n="2"/> |
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19 <pb file="add_6787_f002" o="2" n="3"/> |
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20 <pb file="add_6787_f002v" o="2v" n="4"/> |
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21 <pb file="add_6787_f003" o="3" n="5"/> |
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22 <pb file="add_6787_f003v" o="3v" n="6"/> |
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23 <pb file="add_6787_f004" o="4" n="7"/> |
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24 <pb file="add_6787_f004v" o="4v" n="8"/> |
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25 <pb file="add_6787_f005" o="5" n="9"/> |
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26 <pb file="add_6787_f005v" o="5v" n="10"/> |
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27 <pb file="add_6787_f006" o="6" n="11"/> |
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28 <pb file="add_6787_f006v" o="6v" n="12"/> |
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29 <pb file="add_6787_f007" o="7" n="13"/> |
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30 <pb file="add_6787_f007v" o="7v" n="14"/> |
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31 <pb file="add_6787_f008" o="8" n="15"/> |
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32 <pb file="add_6787_f008v" o="8v" n="16"/> |
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33 <pb file="add_6787_f009" o="9" n="17"/> |
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34 <pb file="add_6787_f009v" o="9v" n="18"/> |
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35 <pb file="add_6787_f010" o="10" n="19"/> |
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36 <pb file="add_6787_f010v" o="10v" n="20"/> |
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37 <pb file="add_6787_f011" o="11" n="21"/> |
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38 <pb file="add_6787_f011v" o="11v" n="22"/> |
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39 <pb file="add_6787_f012" o="12" n="23"/> |
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40 <pb file="add_6787_f012v" o="12v" n="24"/> |
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41 <pb file="add_6787_f013" o="13" n="25"/> |
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42 <pb file="add_6787_f013v" o="13v" n="26"/> |
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43 <pb file="add_6787_f014" o="14" n="27"/> |
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44 <pb file="add_6787_f014v" o="14v" n="28"/> |
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45 <pb file="add_6787_f015" o="15" n="29"/> |
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46 <pb file="add_6787_f015v" o="15v" n="30"/> |
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47 <pb file="add_6787_f016" o="16" n="31"/> |
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48 <pb file="add_6787_f016v" o="16v" n="32"/> |
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49 <pb file="add_6787_f017" o="17" n="33"/> |
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50 <div xml:id="echoid-div1" type="page_commentary" level="2" n="1"> |
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51 <p> |
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52 <s xml:id="echoid-s1" xml:space="preserve">[<emph style="it">Note: |
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53 <p> |
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54 <s xml:id="echoid-s1" xml:space="preserve"> |
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55 The columns show the first, second, third, and fourth differences of an interpolated table |
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56 with constant fourth difference. |
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57 </s> |
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58 </p> |
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59 </emph>] |
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60 <lb/><lb/></s></p></div> |
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61 <head xml:id="echoid-head1" xml:space="preserve" xml:lang="lat"> |
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62 3) Ad calculum sinuum 4<emph style="super">or</emph> differentiarum succedentium |
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63 <lb/>[<emph style="it">tr: |
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64 For the calculation of four successive differences of sines. |
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65 </emph>]<lb/> |
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66 </head> |
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67 <p xml:lang="lat"> |
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68 <s xml:id="echoid-s3" xml:space="preserve"> |
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69 Æquatio ista <lb/> |
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70 examinata <lb/> |
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71 fuit per induct<lb/> |
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72 ione ad aliam <lb/> |
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73 probatam. |
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74 <lb/>[<emph style="it">tr: |
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75 This equation has been examined by the induction used to demonstrate the other. |
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76 </emph>]<lb/> |
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77 </s> |
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78 </p> |
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79 <p xml:lang="lat"> |
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80 <s xml:id="echoid-s4" xml:space="preserve"> |
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81 Reductio <lb/> |
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82 superioris <lb/> |
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83 æquationis <lb/> |
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84 in [¿?]. |
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85 <lb/>[<emph style="it">tr: |
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86 The separation of the above equation into <emph style="super">parts</emph> |
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87 </emph>]<lb/> |
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88 </s> |
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89 </p> |
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90 <pb file="add_6787_f017v" o="17v" n="34"/> |
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91 <pb file="add_6787_f018" o="18" n="35"/> |
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92 <pb file="add_6787_f018v" o="18v" n="36"/> |
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93 <pb file="add_6787_f019" o="19" n="37"/> |
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94 <div xml:id="echoid-div2" type="page_commentary" level="2" n="2"> |
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95 <p> |
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96 <s xml:id="echoid-s5" xml:space="preserve">[<emph style="it">Note: |
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97 <p> |
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98 <s xml:id="echoid-s5" xml:space="preserve"> |
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99 At the top of the page the entries from Add MS 6787, f. 20, now written with common denominator 24. <lb/> |
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100 The lower half of the page shows the first and second differences of the above. |
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101 </s> |
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102 </p> |
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103 </emph>] |
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104 <lb/><lb/></s></p></div> |
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105 <head xml:id="echoid-head2" xml:space="preserve"> |
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106 2) |
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107 </head> |
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108 <p xml:lang="lat"> |
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109 <s xml:id="echoid-s7" xml:space="preserve"> |
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110 Superiorum primariæ differentiæ |
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111 <lb/>[<emph style="it">tr: |
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112 First differences of the above. |
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113 </emph>]<lb/> |
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114 </s> |
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115 <lb/> |
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116 <s xml:id="echoid-s8" xml:space="preserve"> |
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117 Secundariæ differentiæ |
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118 <lb/>[<emph style="it">tr: |
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119 Second differences. |
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120 </emph>]<lb/> |
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121 </s> |
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122 </p> |
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123 <pb file="add_6787_f019v" o="19v" n="38"/> |
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124 <pb file="add_6787_f020" o="20" n="39"/> |
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125 <pb file="add_6787_f020v" o="20v" n="40"/> |
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126 <pb file="add_6787_f021" o="21" n="41"/> |
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127 <pb file="add_6787_f021v" o="21v" n="42"/> |
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128 <pb file="add_6787_f022" o="22" n="43"/> |
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129 <pb file="add_6787_f022v" o="22v" n="44"/> |
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130 <pb file="add_6787_f023" o="23" n="45"/> |
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131 <div xml:id="echoid-div3" type="page_commentary" level="2" n="3"> |
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132 <p> |
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133 <s xml:id="echoid-s9" xml:space="preserve">[<emph style="it">Note: |
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134 <p> |
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135 <s xml:id="echoid-s9" xml:space="preserve"> |
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136 This page contains further work on Viète's statement of 'Syntomon' in Chapter XIX, Proposition 21, of |
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137 <emph style="it">Variorum responsorum liber VIII</emph>. |
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138 </s> |
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139 <lb/> |
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140 <quote xml:lang="lat"> |
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141 Quæ per factionem sub sinibus peripheriarum & adplicationem ad sinum totum exurgunt, |
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142 eadem opere additionis vel subductionis præsto sunt. <lb/> |
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143 Cum duæ peripheriæ angulum acutum componunt, est <lb/> |
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144 Vt sinus totus ad sinum duplum primæ, ita sinus secundæ ad sinum complementi differentia, |
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145 minus sinu complementi composita. |
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146 </quote> |
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147 <lb/> |
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148 <quote> |
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149 What appears from a combination of the sine of the arcs, dividing the sine of the total, |
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150 is also shown by the operations of addition and subtraction. <lb/> |
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151 When two arcs contain acute angles, then as the whole sine is to twice the sine of the first, |
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152 so is the sine of the second to the sum of |
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153 the sine of the complement of the difference minus the sine of the complement of the sum. |
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154 </quote> |
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155 </p> |
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156 </emph>] |
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157 <lb/><lb/></s></p></div> |
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158 <head xml:id="echoid-head3" xml:space="preserve" xml:lang="lat"> |
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159 De 3<emph style="super">o</emph> et 4<emph style="super">o</emph> casu |
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160 <foreign xml:lang="gre">ton syntomon</foreign> (1. |
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161 <lb/>[<emph style="it">tr: |
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162 On cases 3 and 4 of Syntomon |
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163 </emph>]<lb/> |
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164 </head> |
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165 <p xml:lang="lat"> |
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166 <s xml:id="echoid-s11" xml:space="preserve"> |
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167 Etsi duo priores casus <lb/> |
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168 sufficiunt ad operationes: <lb/> |
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169 duo tamen sequentes ad <lb/> |
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170 argumentationes sunt ali-<lb/> |
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171 quando necessarij. |
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172 <lb/>[<emph style="it">tr: |
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173 Although the two first cases suffice for working, |
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174 nevertheless the two following arguments are sometimes necessary. |
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175 </emph>]<lb/> |
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176 </s> |
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177 </p> |
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178 <p xml:lang="lat"> |
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179 <s xml:id="echoid-s12" xml:space="preserve"> |
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180 3<emph style="super">us</emph> casus est, quando unus <lb/> |
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181 datorum arcuum sit maior <lb/> |
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182 quadrante; et differentia sit <lb/> |
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183 etiam quadrante maior. |
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184 <lb/>[<emph style="it">tr: |
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185 The 3rd case is when one of the given arcs is greater than a quadrant, and the difference is also greater than a quadrant. |
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186 </emph>]<lb/> |
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187 </s> |
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188 </p> |
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189 <p xml:lang="lat"> |
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190 <s xml:id="echoid-s13" xml:space="preserve"> |
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191 4<emph style="super">us</emph> casus est, quando unus datorum <lb/> |
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192 arcuum sit maior quadrante; <lb/> |
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193 sed differentia sit quadrante minore. |
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194 <lb/>[<emph style="it">tr: |
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195 The 4th case is when one of the given arcs is greater than a quadrant, but the difference is less. |
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196 </emph>]<lb/> |
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197 </s> |
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198 </p> |
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199 <pb file="add_6787_f023v" o="23v" n="46"/> |
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200 <pb file="add_6787_f024" o="24" n="47"/> |
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201 <div xml:id="echoid-div4" type="page_commentary" level="2" n="4"> |
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202 <p> |
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203 <s xml:id="echoid-s14" xml:space="preserve">[<emph style="it">Note: |
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204 <p> |
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205 <s xml:id="echoid-s14" xml:space="preserve"> |
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206 This page contains further work on Viète's statement of 'Syntomon' in Chapter XIX, Proposition 21, of |
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207 <emph style="it">Variorum responsorum liber VIII</emph>. |
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208 </s> |
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209 </p> |
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210 </emph>] |
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211 <lb/><lb/></s></p></div> |
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212 <head xml:id="echoid-head4" xml:space="preserve" xml:lang="lat"> |
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213 Ad triangula sphærica obliquangula Vietæ (2. |
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214 <lb/>[<emph style="it">tr: |
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215 On Viète's obtuse-angle spherical triangles. |
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216 </emph>]<lb/> |
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217 </head> |
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218 <p xml:lang="lat"> |
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219 <s xml:id="echoid-s16" xml:space="preserve"> |
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220 Sint arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math>, et, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math>, sigillatim minores quadrante: <lb/> |
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221 Dico quod: <lb/>[...]<lb/> hoc est sinuui comple-<lb/> |
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222 menti differentiæ <lb/> |
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223 arcuum datorum. |
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224 <lb/>[<emph style="it">tr: |
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225 Let the arcs <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math> each be less than a quadrant. <lb/> |
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226 I say that <lb/>[...]<lb/> this is the sine of the complement of the difference of the given arcs. |
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227 </emph>]<lb/> |
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228 </s> |
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229 </p> |
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230 <p xml:lang="lat"> |
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231 <s xml:id="echoid-s17" xml:space="preserve"> |
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232 Ad sinus magisterij demonstrationem, dico quod: |
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233 <lb/>[<emph style="it">tr: |
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234 For demonstrating the doctrine of sines, I say that: |
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235 </emph>]<lb/> |
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236 </s> |
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237 </p> |
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238 <p xml:lang="lat"> |
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239 <s xml:id="echoid-s18" xml:space="preserve"> |
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240 Quæ demonstrari oportuit. |
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241 <lb/>[<emph style="it">tr: |
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242 Which was to be demonstrated. |
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243 </emph>]<lb/> |
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244 </s> |
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245 </p> |
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246 <pb file="add_6787_f024v" o="24v" n="48"/> |
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247 <pb file="add_6787_f025" o="25" n="49"/> |
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248 <div xml:id="echoid-div5" type="page_commentary" level="2" n="5"> |
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249 <p> |
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250 <s xml:id="echoid-s19" xml:space="preserve">[<emph style="it">Note: |
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251 <p> |
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252 <s xml:id="echoid-s19" xml:space="preserve"> |
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253 This page contains further work on Viète's statement of 'Syntomon' in Chapter XIX, Proposition 21, of |
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254 <emph style="it">Variorum responsorum liber VIII</emph>. |
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255 </s> |
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256 </p> |
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257 </emph>] |
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258 <lb/><lb/></s></p></div> |
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259 <head xml:id="echoid-head5" xml:space="preserve" xml:lang="lat"> |
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260 (3. |
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261 </head> |
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262 <p xml:lang="lat"> |
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263 <s xml:id="echoid-s21" xml:space="preserve"> |
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264 Aliter. <lb/> |
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265 In diagrammatis syntomi |
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266 (<emph style="super">In</emph> utriusque casibus, 1<emph style="super">o</emph> et 2<emph style="super">o</emph>) <lb/> |
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267 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math>, secat, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>b</mi></mstyle></math>, in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math>. <lb/> |
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268 fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi></mstyle></math> parallela <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>n</mi></mstyle></math>. <lb/> |
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269 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi></mstyle></math> secabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>g</mi></mstyle></math> in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. |
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270 <lb/>[<emph style="it">tr: |
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271 Another way, <lb/> |
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272 In the diagram for syntomon, (in either case 1 or 2) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math> cuts <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>b</mi></mstyle></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math>. |
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273 Make <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi></mstyle></math> parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>n</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>z</mi></mstyle></math> will cut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>g</mi></mstyle></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. |
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274 </emph>]<lb/> |
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275 </s> |
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276 </p> |
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277 <p xml:lang="lat"> |
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278 <s xml:id="echoid-s22" xml:space="preserve"> |
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279 1<emph style="super">m</emph> Dico quod: <lb/>[...]<lb/> hoc est sinuui <lb/> |
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280 complementi <reg norm="differentiaæ" type="abbr">dræ</reg> <lb/> |
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281 arcuum datorum. |
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282 <lb/>[<emph style="it">tr: |
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283 1st, I say that this is the sine of the complement of the difference of the given arcs. |
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284 </emph>]<lb/> |
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285 </s> |
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286 </p> |
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287 <p xml:lang="lat"> |
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288 <s xml:id="echoid-s23" xml:space="preserve"> |
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289 Quod fiat demonstrandum. |
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290 <lb/>[<emph style="it">tr: |
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291 Which was to be demonstrated. </emph>]<lb/> |
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292 </s> |
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293 </p> |
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294 <p xml:lang="lat"> |
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295 <s xml:id="echoid-s24" xml:space="preserve"> |
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296 Sequitur etiam per 1<emph style="super">m</emph>: quod: <lb/> |
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297 in utroque <lb/> |
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298 casu. |
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299 <lb/>[<emph style="it">tr: |
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300 There follows from the first, because in either case: |
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301 </emph>]<lb/> |
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302 </s> |
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303 </p> |
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304 <pb file="add_6787_f025v" o="25v" n="50"/> |
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305 <pb file="add_6787_f026" o="26" n="51"/> |
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306 <div xml:id="echoid-div6" type="page_commentary" level="2" n="6"> |
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307 <p> |
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308 <s xml:id="echoid-s25" xml:space="preserve">[<emph style="it">Note: |
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309 <p> |
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310 <s xml:id="echoid-s25" xml:space="preserve"> |
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311 This page contains further work on spherical triangles. |
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312 </s> |
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313 </p> |
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314 </emph>] |
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315 <lb/><lb/></s></p></div> |
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316 <head xml:id="echoid-head6" xml:space="preserve" xml:lang="lat"> |
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317 (4. |
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318 </head> |
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319 <p xml:lang="lat"> |
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320 <s xml:id="echoid-s27" xml:space="preserve"> |
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321 In utriusque casibus, 3<emph style="super">o</emph> et 4<emph style="super">o</emph> |
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322 <foreign xml:lang="gre">ton syntomon</foreign> <lb/> |
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323 1<emph style="super">m</emph>) Dico quod: |
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324 <lb/>[<emph style="it">tr: |
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325 In either the 3rd or 4th case of Syntomon <lb/> |
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326 1) I say that: |
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327 </emph>]<lb/> |
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328 </s> |
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329 </p> |
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330 <p xml:lang="lat"> |
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331 <s xml:id="echoid-s28" xml:space="preserve"> |
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332 2<emph style="super">m</emph>) Dico quod in 3<emph style="super">m</emph> casu: |
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333 <lb/>[<emph style="it">tr: |
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334 2) I say that in the 3rd case: |
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335 </emph>]<lb/> |
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336 </s> |
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337 </p> |
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338 <p xml:lang="lat"> |
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339 <s xml:id="echoid-s29" xml:space="preserve"> |
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340 3<emph style="super">m</emph>) Dico quod in 4<emph style="super">m</emph> casu: |
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341 <lb/>[<emph style="it">tr: |
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342 3) I say that in the 4th case: |
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343 </emph>]<lb/> |
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344 </s> |
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345 </p> |
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346 <p xml:lang="lat"> |
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347 <s xml:id="echoid-s30" xml:space="preserve"> |
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348 Sequitur etiam per 1<emph style="super">m</emph>: quod: <lb/> |
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349 in utroque casu, 3<emph style="super">o</emph> et 4<emph style="super">o</emph>. |
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350 <lb/>[<emph style="it">tr: |
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351 There follows also from the first, because in either the 3rd or 4th case: |
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352 </emph>]<lb/> |
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353 </s> |
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354 </p> |
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355 <pb file="add_6787_f026v" o="26v" n="52"/> |
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356 <pb file="add_6787_f027" o="27" n="53"/> |
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357 <div xml:id="echoid-div7" type="page_commentary" level="2" n="7"> |
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358 <p> |
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359 <s xml:id="echoid-s31" xml:space="preserve">[<emph style="it">Note: |
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360 <p> |
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361 <s xml:id="echoid-s31" xml:space="preserve"> |
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362 This page contains further work on spherical triangles. <lb/> |
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363 There are references to both Regiomontanus and Clavius, |
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364 who both gave a version of the theorem given here. <lb/> |
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365 The reference to Regiomontanus is to his <emph style="it">De triangulis omnimodis libri quinque</emph> (1533), |
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366 Book V, Proposition II. (For another reference to the same proposition, see also Add MS 6782, f. 490.) |
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367 </s> |
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368 <lb/> |
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369 <quote xml:lang="lat"> |
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370 V.II In omni triangulo sphaerali ex arcubus circulorum magnorum constante, |
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371 proportio sinus uersi anguli cuislibet ad differentiam duorum sinuum uersorum, |
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372 quorum unus est lateris eum angulum subtendentis: |
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373 alius uerò differentiae duorum arcuum ipsi angulo circumiacentium |
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374 est tanquam proportio quadrati sinus recti totius ad id, |
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375 quod sub sinibus arcuum dicto angulo circumpositorum continetur rectangulum |
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376 </quote> |
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377 <lb/> |
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378 <quote> |
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379 In all spherical triangles composed from great arcs of circles, the ratio of the versed sine of any angle |
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380 to the difference of two versed sines, one of which is the side subtending the angle, |
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381 the other the difference of the two arcs adjacent to the angle, |
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382 is the proportion of the the square of the whole sine |
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383 to the product of the sines of the surrounding arcs by which the said angle is contained. |
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384 </quote> |
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385 <lb/> |
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386 <s xml:id="echoid-s32" xml:space="preserve"> |
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387 The reference to Clavius is to his <emph style="it">Triangula sphærica</emph> in |
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388 <emph style="it">Triangula rectilinea, atque sphaerica</emph> (1586). |
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389 Proposition 58, on page 445. |
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390 </s> |
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391 <lb/> |
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392 <quote xml:lang="lat"> |
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393 Theorema 56, Propositio 58. <lb/> |
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394 In omni triangulo sphærico, cuius duo arcus sint inæquales; |
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395 quadratum sinus totius ad rectangulum sub sinubus rectis duorum arcuum inæqulium contentum, |
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396 eandem proportionem habet, quam sinus versus anguli a dictis arcubus comprehensi |
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397 ad differentiam duorum sinuum versorum, quorum vnus differentiæ eorundem arcuum debetur, |
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398 alter vero tertio arcui, qui prædicto angulo oppostitus est, respondet. |
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399 </quote> |
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400 <lb/> |
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401 <quote> |
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402 In all spherical triangles, whose two arcs are unequal, |
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403 the square of the whole sine to the product of the sines of the two unequal arcs |
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404 is in the same ratio as the versed sine of the angle between the said arcs |
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405 to the difference of two versed sines, one of which is of the difference of the arcs, |
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406 the other corresponding to the third arc, which is opposite the aforesaid angle. |
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407 </quote> |
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408 <lb/> |
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409 <s xml:id="echoid-s33" xml:space="preserve"> |
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410 Harrot translates Clavius's statement into symbols for the particular triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>a</mi><mi>b</mi></mstyle></math> shown in his diagram. |
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411 He then goes on to prove that the versed sine of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi><mo>-</mo><mi>d</mi><mi>b</mi></mstyle></math> is greater than the versed sine of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. <lb/> |
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412 For another version of this page, see Add MS 6787, f. 51. |
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413 </s> |
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414 </p> |
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415 </emph>] |
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416 <lb/><lb/></s></p></div> |
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417 <head xml:id="echoid-head7" xml:space="preserve" xml:lang="lat"> |
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418 (5. <lb/> |
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419 Analogia per sinus versos, et universalis ad triangula <lb/> |
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420 sphærica cuiuscunque conditionis. |
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421 <lb/>[<emph style="it">tr: |
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422 Ratio by versed sines, and generally for a spherical trianlge under any conditions. |
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423 </emph>]<lb/> |
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424 </head> |
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425 <p xml:lang="lat"> |
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426 <s xml:id="echoid-s35" xml:space="preserve"> |
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427 Demonstratur a Regiomontano <lb/> |
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428 lib. 5<emph style="super">o</emph>. pr. 2, de triang. <lb/> |
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429 A clavio pr. 58. de sphæricis <lb/> |
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430 Ab alijs Trigonistis. <lb/> |
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431 Et a nobis alibi in notis. |
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432 <lb/>[<emph style="it">tr: |
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433 Demonstrated by Regiomontanus in <emph style="it">De triangulis</emph>, Book 5, Proposition 2. <lb/> |
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434 by Clavius in Proposition 58 of <emph style="it">De sphæricis triangulis</emph> <lb/> |
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435 By other triangulists. <lb/> |
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436 And by me elsewhere in notation. |
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437 </emph>]<lb/> |
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438 [<emph style="it">Note: |
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439 The page 'elsewhere' referred to here is probably Add MS 6787, f. 27. |
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440 </emph>]<lb/> |
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441 </s> |
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442 </p> |
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443 <p xml:lang="lat"> |
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444 <s xml:id="echoid-s36" xml:space="preserve"> |
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445 Dico quod: <lb/>[...]<lb/> (<foreign xml:lang="gre">catolicos</foreign>) |
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446 <lb/>[<emph style="it">tr: |
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447 I say that <lb/>[...]<lb/> (generally) |
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448 </emph>]<lb/> |
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449 </s> |
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450 </p> |
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451 <p xml:lang="lat"> |
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452 <s xml:id="echoid-s37" xml:space="preserve"> |
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453 Consectarium <lb/> |
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454 ponatur quod: <lb/> |
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455 In triangulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi><mi>b</mi></mstyle></math>, datis duobus lateribus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math>; cum angulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> <lb/> |
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456 queratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. <lb/> |
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457 per superiorem analogiam, sit <emph style="super">data</emph> quarta proportionalis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. <lb/> |
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458 <lb/>[...]<lb/> <lb/> |
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459 datur igitur, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. |
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460 <lb/>[<emph style="it">tr: |
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461 Consequence <lb/> |
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462 It is supposed that, in triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi><mi>b</mi></mstyle></math>, from given two sides <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math> with the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, |
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463 there is sought <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. <lb/> |
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464 by the above ratio, let the given fourth proportional be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math> <lb/> |
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465 <omissions/> <lb/> |
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466 therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> is given |
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467 </emph>]<lb/> |
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468 </s> |
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469 </p> |
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470 <pb file="add_6787_f027v" o="27v" n="54"/> |
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471 <pb file="add_6787_f028" o="28" n="55"/> |
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472 <div xml:id="echoid-div8" type="page_commentary" level="2" n="8"> |
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473 <p> |
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474 <s xml:id="echoid-s38" xml:space="preserve">[<emph style="it">Note: |
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475 <p> |
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476 <s xml:id="echoid-s38" xml:space="preserve"> |
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477 Further work on spherical triangles. |
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478 </s> |
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479 </p> |
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480 </emph>] |
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481 <lb/><lb/></s></p></div> |
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482 <head xml:id="echoid-head8" xml:space="preserve" xml:lang="lat"> |
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483 (6. <lb/> |
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484 Investigatio analogiæ <lb/> |
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485 Vietanæ |
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486 <lb/>[<emph style="it">tr: |
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487 Investigation of Viète's ratios. |
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488 </emph>]<lb/> |
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489 </head> |
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490 <p xml:lang="lat"> |
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491 <s xml:id="echoid-s40" xml:space="preserve"> |
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492 Hoc est magisterium. |
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493 <lb/>[<emph style="it">tr: |
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494 This is the rule. |
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495 </emph>]<lb/> |
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496 </s> |
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497 </p> |
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498 <p xml:lang="lat"> |
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499 <s xml:id="echoid-s41" xml:space="preserve"> |
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500 Ut Vieta <lb/> |
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501 pag. 47b. |
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502 <lb/>[<emph style="it">tr: |
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503 As Viète page 47v. |
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504 </emph>]<lb/> |
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505 </s> |
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506 </p> |
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507 <p xml:lang="lat"> |
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508 <s xml:id="echoid-s42" xml:space="preserve"> |
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509 Ut Vieta <lb/> |
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510 pag. 35b. |
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511 <lb/>[<emph style="it">tr: |
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512 As Viete, page 35v. |
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513 </emph>]<lb/> |
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514 </s> |
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515 </p> |
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516 <p xml:lang="lat"> |
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517 <s xml:id="echoid-s43" xml:space="preserve"> |
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518 Conditiones <lb/> |
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519 alterius trianguli <lb/> |
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520 contrariæ, sub eadem <lb/> |
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521 analogia. |
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522 <lb/>[<emph style="it">tr: |
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523 Conditions for another triangle, conversely, by the same ratio. |
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524 </emph>]<lb/> |
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525 </s> |
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526 </p> |
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527 <p xml:lang="lat"> |
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528 <s xml:id="echoid-s44" xml:space="preserve"> |
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529 Nota. Etsi signum (<) ponatur sub <lb/>[...]<lb/>, intelligitur quod, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo><</mo><mn>9</mn><mn>0</mn></mstyle></math>. <lb/> |
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530 Ita signum (>) sub (<lb/>[...]<lb/>) denotat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>></mo><mn>9</mn><mn>0</mn></mstyle></math>. <lb/> |
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531 Istud signum (<lb/>[...]<lb/>), denotat unum latus maius altera minus 90 &c. <lb/> |
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532 Et in alijs locis, (<lb/>[...]<lb/>), utrinque minus, utrinque maius 90. |
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533 <lb/>[<emph style="it">tr: |
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534 Note. If this sign (<) is put below <lb/>[...]<lb/>, it is to be understood that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo><</mo><mn>9</mn><mn>0</mn></mstyle></math>. <lb/> |
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535 This sign (>) below (<lb/>[...]<lb/>) denotes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo><</mo><mn>9</mn><mn>0</mn></mstyle></math>. <lb/> |
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536 This sign (<lb/>[...]<lb/>) denotes one side is greater than the other minus 90. <lb/> |
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537 An in other places, (<omision/>) eother less than or greater than 90. |
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538 </emph>]<lb/> |
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539 </s> |
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540 </p> |
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541 <pb file="add_6787_f028v" o="28v" n="56"/> |
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542 <pb file="add_6787_f029" o="29" n="57"/> |
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543 <div xml:id="echoid-div9" type="page_commentary" level="2" n="9"> |
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544 <p> |
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545 <s xml:id="echoid-s45" xml:space="preserve">[<emph style="it">Note: |
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546 <p> |
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547 <s xml:id="echoid-s45" xml:space="preserve"> |
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548 Further work on spherical triangles. |
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549 </s> |
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550 </p> |
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551 </emph>] |
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552 <lb/><lb/></s></p></div> |
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553 <head xml:id="echoid-head9" xml:space="preserve" xml:lang="lat"> |
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554 (7. |
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555 </head> |
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556 <p xml:lang="lat"> |
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557 <s xml:id="echoid-s47" xml:space="preserve"> |
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558 Investigatio trium analogiarum <lb/> |
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559 sive casuum. <lb/> |
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560 ex positis sequentibus |
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561 <lb/>[<emph style="it">tr: |
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562 Investigation of three ratios or cases, supposing the following. |
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563 </emph>]<lb/> |
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564 </s> |
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565 </p> |
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566 <pb file="add_6787_f029v" o="29v" n="58"/> |
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567 <pb file="add_6787_f030" o="30" n="59"/> |
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568 <pb file="add_6787_f030v" o="30v" n="60"/> |
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569 <pb file="add_6787_f031" o="31" n="61"/> |
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570 <div xml:id="echoid-div10" type="page_commentary" level="2" n="10"> |
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571 <p> |
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572 <s xml:id="echoid-s48" xml:space="preserve">[<emph style="it">Note: |
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573 <p> |
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574 <s xml:id="echoid-s48" xml:space="preserve"> |
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575 The reference on this page is to Proposition 21 from Chapter 19 of Viète's |
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576 <emph style="it">Variorum responsorum liber VIII</emph> (1593). |
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577 </s> |
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578 <lb/> |
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579 <quote xml:lang="lat"> |
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580 XXI. <lb/> |
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581 Trianguli cujuslibet sphærici. <lb/> |
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582 Datis duobus lateribus, & angulo cui unum ex illis lateribus opponitur, |
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583 datur angulus cui alterum datorum laterum opponitur. <lb/> |
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584 Vel, <lb/> |
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585 Datis duobs angulis, & latere quod alteri datorum angulorum opponitur, |
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586 datur latus reliquo oppositum. |
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587 </quote> |
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588 <lb/> |
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589 <quote> |
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590 Given two sides and the angle opposite one of those sides, the angle opposite the other is known. <lb/> |
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591 Or, <lb/> |
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592 Given two angles, and the side opposite one of the given angles, the side opposite the other is known. |
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593 </quote> |
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594 <lb/> |
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595 <s xml:id="echoid-s49" xml:space="preserve"> |
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596 Immediately after the statement of the proposition, Viète gave the following statement, |
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597 under the heading Syntomon. |
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598 </s> |
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599 <lb/> |
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600 <quote xml:lang="lat"> |
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601 Quæ per factionem sub sinibus peripheriarum & adplicationem ad sinum totum exurgunt, |
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602 eadem opere additionis vel subductionis præsto sunt. <lb/> |
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603 Cum duæ peripheriæ angulum acutum componunt, est <lb/> |
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604 Vt sinus totus ad sinum duplum primæ, ita sinus secundæ ad sinum complementi differentia, |
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605 minus sinu complementi composita. |
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606 </quote> |
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607 <lb/> |
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608 <quote> |
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609 What appears from a combination of the sine of the arcs, dividing the sine of the total, |
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610 is also shown by the operations of addition and subtraction. <lb/> |
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611 When two arcs contain acute angles, then as the whole sine is to twice the sine of the first, |
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612 so is the sine of the second to the sum of |
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613 the sine of the complement of the difference minus the sine of the complement of the sum. |
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614 </quote> |
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615 <lb/> |
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616 <s xml:id="echoid-s50" xml:space="preserve"> |
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617 In modern notation this statement may be written as: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>:</mo><mrow><mn>2</mn><mspace width="0.167em"/></mrow><mrow><mi>sin</mi><mrow><mi>a</mi></mrow></mrow><mo>=</mo><mrow><mi>sin</mi><mrow><mi>b</mi></mrow></mrow><mo>:</mo><mrow><mi>cos</mi><mrow><mi>a</mi><mo>-</mo><mi>b</mi></mrow></mrow><mo>+</mo><mrow><mi>cos</mi><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mrow></mstyle></math>. |
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618 This is the ratio Harriot has written nex to diagram 1, where both angles are acute. |
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619 The other diagrams are for cases where one or both the angles are obtuse. |
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620 </s> |
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621 </p> |
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622 </emph>] |
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623 <lb/><lb/></s></p></div> |
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624 <head xml:id="echoid-head10" xml:space="preserve" xml:lang="lat"> |
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625 Vieta lib. 8. resp. |
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626 pag. 39. <lb/> |
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627 <foreign xml:lang="gre">Syntomon</foreign> |
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628 <lb/>[<emph style="it">tr: |
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629 Viète, Responsorum liber VIII. <lb/> |
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630 Syntomon |
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631 </emph>]<lb/> |
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632 </head> |
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633 <p xml:lang="lat"> |
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634 <s xml:id="echoid-s52" xml:space="preserve"> |
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635 [???] in alia charta |
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636 <lb/>[<emph style="it">tr: |
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637 [???] in another sheet |
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638 </emph>]<lb/> |
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639 </s> |
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640 </p> |
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641 <p xml:lang="lat"> |
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642 <s xml:id="echoid-s53" xml:space="preserve"> |
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643 Quæ per factionem sub sinibus peripherieriarum et adplicationem ad sinum totum exurgunt, |
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644 eadem opere additionis vel subductionis præsto sunt. |
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645 <lb/>[<emph style="it">tr: |
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646 What appears from a combination of the sine of the arcs, dividing the sine of the total, |
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647 is also shown by the operations of addition and subtraction. |
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648 </emph>]<lb/> |
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649 </s> |
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650 </p> |
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651 <p xml:lang="lat"> |
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652 <s xml:id="echoid-s54" xml:space="preserve"> |
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653 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, una peripheria <lb/> |
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654 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, altera peripheria <lb/> |
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655 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math>, differentia <lb/> |
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656 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, aggregatum &c. |
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657 <lb/>[<emph style="it">tr: |
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658 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> is one arc, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> the other. <lb/> |
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659 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> is the difference, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math> the sum. |
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660 </emph>]<lb/> |
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661 </s> |
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662 </p> |
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663 <p xml:lang="lat"> |
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664 <s xml:id="echoid-s55" xml:space="preserve"> |
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665 Hæc quarta analogia est re eadem <lb/> |
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666 cum secunda. <lb/> |
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667 porro, prima et tertia analogiæ <lb/> |
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668 reducuntur ad unam si quartus <lb/> |
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669 terminus ita notetur. |
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670 <lb/>[<emph style="it">tr: |
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671 This fourth ratio is the same thing as the second. <lb/> |
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672 Further, the first and third ratios are reduced to one if the fourth term is written thus. |
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673 </emph>]<lb/> |
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674 [<emph style="it">Note: |
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675 Here the symbols that looks like an equals sign is to be read as a minus sign, |
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676 where the smaller quantity is always understood to be subtracted from the larger. |
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677 </emph>]<lb/> |
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678 </s> |
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679 </p> |
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680 <p xml:lang="lat"> |
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681 <s xml:id="echoid-s56" xml:space="preserve"> |
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682 Nota <lb/> |
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683 Quando una peripheria est maior quadranti <lb/> |
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684 ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> est <emph style="super">in</emph> |
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685 3,<emph style="super">a</emph> et 4,<emph style="super">a</emph> diagrammati; summatur <lb/> |
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686 eius residuum ad semicirculum. Et tum <lb/> |
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687 operatio erit secundum primum vel secundum casum. <lb/> |
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688 Quare hoc modo sunt duo tantummodo <lb/> |
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689 casus. |
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690 <lb/>[<emph style="it">tr: |
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691 Note. <lb/> |
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692 When one arc is greater than a quadrant, as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> is in the 3rd and 4th diagrams, |
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693 there are taken their residuals from a semicircle. |
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694 And then the operation witll be as the first or second case. <lb/> |
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695 Therefore by this method there are only two cases. |
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696 </emph>]<lb/> |
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697 </s> |
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698 </p> |
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699 <pb file="add_6787_f031v" o="31v" n="62"/> |
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700 <pb file="add_6787_f032" o="32" n="63"/> |
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701 <div xml:id="echoid-div11" type="page_commentary" level="2" n="11"> |
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702 <p> |
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703 <s xml:id="echoid-s57" xml:space="preserve">[<emph style="it">Note: |
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704 <p> |
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705 <s xml:id="echoid-s57" xml:space="preserve"> |
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706 This page continues Harriot's work from Add MS 6787, f. 61, |
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707 on Viète's statement of 'Syntomon'. |
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708 </s> |
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709 </p> |
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710 </emph>] |
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711 <lb/><lb/></s></p></div> |
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712 <head xml:id="echoid-head11" xml:space="preserve" xml:lang="lat"> |
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713 2) syntomon |
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714 <lb/>[<emph style="it">tr: |
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715 2) Syntomon |
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716 </emph>]<lb/> |
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717 </head> |
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718 <p xml:lang="lat"> |
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719 <s xml:id="echoid-s59" xml:space="preserve"> |
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720 primus casus. quando <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>a</mi><mo><</mo><mn>9</mn><mn>0</mn></mstyle></math>. <lb/> |
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721 Ponatur analogia |
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722 <lb/>[<emph style="it">tr: |
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723 First case, when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>a</mi><mo><</mo><mn>9</mn><mn>0</mn></mstyle></math>. <lb/> |
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724 There is put the ratio: |
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725 </emph>]<lb/> |
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726 </s> |
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727 </p> |
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728 <p xml:lang="lat"> |
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729 <s xml:id="echoid-s60" xml:space="preserve"> |
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730 sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> maior arcus <lb/> |
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731 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> minor <lb/> |
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732 sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math> <lb/> |
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733 ideo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> differentia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mi>b</mi></mstyle></math>. <lb/> |
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734 sit etiam, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>+</mo><mi>a</mi><mi>c</mi><mo><</mo><mn>9</mn><mn>0</mn></mstyle></math>: pro 1<emph style="super">o</emph> casu. <lb/> |
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735 Dico quod <lb/> |
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736 vel sub hæc speciæ |
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737 <lb/>[<emph style="it">tr: |
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738 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> be the greater arc, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> the lesser, and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>. <lb/> |
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739 Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> is the difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mi>b</mi></mstyle></math>. <lb/> |
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740 Let also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>+</mo><mi>a</mi><mi>c</mi><mo><</mo><mn>9</mn><mn>0</mn></mstyle></math> for the first case. <lb/> |
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741 I say that: <lb/> |
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742 Or in this general form: |
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743 </emph>]<lb/> |
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744 </s> |
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745 </p> |
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746 <p xml:lang="lat"> |
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747 <s xml:id="echoid-s61" xml:space="preserve"> |
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748 nota <lb/> |
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749 complementum differentiæ <lb/> |
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750 complementum aggregati |
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751 <lb/>[<emph style="it">tr: |
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752 Note. <lb/> |
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753 Complement of the difference. <lb/> |
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754 Complement of the sum. |
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755 </emph>]<lb/> |
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756 </s> |
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757 </p> |
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758 <p xml:lang="lat"> |
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759 <s xml:id="echoid-s62" xml:space="preserve"> |
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760 Secundus casus est quando <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>+</mo><mi>a</mi><mi>b</mi><mo>></mo><mn>9</mn><mn>0</mn></mstyle></math>. <lb/> |
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761 <lb/>[...]<lb/> <lb/> |
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762 Quod etiam demonstrandum fuit. |
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763 <lb/>[<emph style="it">tr: |
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764 The second case is wehn <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>+</mo><mi>a</mi><mi>b</mi><mo>></mo><mn>9</mn><mn>0</mn></mstyle></math>. <lb/> |
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765 <lb/>[...]<lb/> <lb/> |
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766 Which was also to be demonstrated. |
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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_6787_f032v" o="32v" n="64"/> |
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771 <pb file="add_6787_f033" o="33" n="65"/> |
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772 <div xml:id="echoid-div12" type="page_commentary" level="2" n="12"> |
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773 <p> |
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774 <s xml:id="echoid-s63" xml:space="preserve">[<emph style="it">Note: |
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775 <p> |
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776 <s xml:id="echoid-s63" xml:space="preserve"> |
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777 Further work on spherical triangles. |
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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 <head xml:id="echoid-head12" xml:space="preserve" xml:lang="lat"> |
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783 (8. |
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784 </head> |
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785 <p xml:lang="lat"> |
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786 <s xml:id="echoid-s65" xml:space="preserve"> |
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787 Investigatio trium aliarum <lb/> |
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788 analogiarum sive casuum. <lb/> |
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789 ex positis sequentibus |
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790 <lb/>[<emph style="it">tr: |
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791 Investigation of three other ratios or cases, supposing the following. |
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792 </emph>]<lb/> |
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793 </s> |
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794 </p> |
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795 <p xml:lang="lat"> |
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796 <s xml:id="echoid-s66" xml:space="preserve"> |
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797 per antithesin <lb/> |
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798 vel conversione <lb/> |
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799 signorum: <lb/> |
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800 istæ æquationes <lb/> |
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801 sunt eadem <lb/> |
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802 cum illis in <lb/> |
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803 antecedente <lb/> |
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804 charta. |
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805 <lb/>[<emph style="it">tr: |
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806 By antithesis or change of sign, these equations are the same as those in the preceding sheet. |
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807 </emph>]<lb/> |
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808 [<emph style="it">Note: |
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809 The preceding sheet was Add MS 6787, f. 29. |
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810 </emph>]<lb/> |
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811 </s> |
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812 </p> |
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813 <p xml:lang="lat"> |
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814 <s xml:id="echoid-s67" xml:space="preserve"> |
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815 Ergo analogiæ sunt etiam eædem ut antea, nisi quod <lb/> |
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816 conditiones sunt contrariæ. |
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817 <lb/>[<emph style="it">tr: |
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818 Therefore the ratios are also the samse as before, unless the conditions are opposite. |
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819 </emph>]<lb/> |
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820 </s> |
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821 </p> |
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822 <pb file="add_6787_f033v" o="33v" n="66"/> |
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823 <pb file="add_6787_f034" o="34" n="67"/> |
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824 <div xml:id="echoid-div13" type="page_commentary" level="2" n="13"> |
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825 <p> |
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826 <s xml:id="echoid-s68" xml:space="preserve">[<emph style="it">Note: |
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827 <p> |
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828 <s xml:id="echoid-s68" xml:space="preserve"> |
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829 Further work on spherical triangles. <lb/> |
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830 As on Add MS 6787, f. 27, there is reference to Clavius, |
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831 <emph style="it">Triangula rectilinea, atque sphaerica</emph> (1586), |
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832 this time to Proposition 27. |
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833 </s> |
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834 </p> |
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835 </emph>] |
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836 <lb/><lb/></s></p></div> |
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837 <head xml:id="echoid-head13" xml:space="preserve" xml:lang="lat"> |
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838 (9. <lb/> |
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839 Casuum Limitatio cum designatione adcomodum |
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840 <lb/>[<emph style="it">tr: |
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841 Determination of cases with a useful specification |
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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-s70" xml:space="preserve"> |
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846 Animadvertendum quod in superioribus investigationibus analogiarum <lb/> |
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847 sive casuum, ubi latera <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math> signantur < <; debeat etiam <lb/> |
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848 intelligi signata > >. Quoniam si sint eiusdem affectionis <lb/> |
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849 sive utrique minora quadrantibus, sive maiora; non variatur <lb/> |
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850 inde illatio neque casus. <lb/> |
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851 Hinc in 6 notatis casibus positi trianguli, sunt 9 variationes <lb/> |
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852 signorum ut sequitur. |
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853 <lb/>[<emph style="it">tr: |
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854 It is to be noted that in the above investigations of ratios, or cases, |
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855 where the sides <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>d</mi><mi>b</mi></mstyle></math> are marked with <, <, respectively, |
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856 it is also to be understood that they could be marked with >, >, respectively. |
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857 Because they have the same relationship, whether both less than a quadrant, or greater; |
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858 therefore the result does not vary, nor the cases. <lb/> |
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859 Here in the 6 denoted cases of the supposed triangle, there are 9 variations. |
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860 </emph>]<lb/> |
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861 </s> |
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862 </p> |
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863 <p xml:lang="lat"> |
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864 <s xml:id="echoid-s71" xml:space="preserve"> |
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865 Sunt præter illas novem, tres aliæ variationes; et non dantur <lb/> |
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866 plures: sed istæ sunt impossibiles (hypostatice). |
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867 <lb/>[<emph style="it">tr: |
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868 Besides those nine, there are three other variations; more are not given, but these are impossible. |
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869 </emph>]<lb/> |
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870 </s> |
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871 </p> |
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872 <p xml:lang="lat"> |
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873 <s xml:id="echoid-s72" xml:space="preserve"> |
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874 Una <emph style="super">et ultima</emph> impossibilium probatur per Clavium pro: 27 de Sphæricis. <lb/> |
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875 et a me magis perspicus in notis de conversione triangulorum: <lb/> |
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876 reliquæ duæ conseqununtur per paraplerosin. |
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877 <lb/>[<emph style="it">tr: |
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878 One, the final impossibility, is proved by Clavus in Proposition 27 of De sphærica; |
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879 and by me more clearly in notation in the conversion of triangles; the remaining two follow by paraplerosis. |
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880 </emph>]<lb/> |
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881 </s> |
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882 </p> |
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883 <p xml:lang="lat"> |
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884 <s xml:id="echoid-s73" xml:space="preserve"> |
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885 Alia designatio analogiarum et casuum, <lb/> |
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886 usui magi adcommoda. |
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887 <lb/>[<emph style="it">tr: |
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888 Another specification of ratios and cases, more convenient in practice. |
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889 </emph>]<lb/> |
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890 </s> |
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891 </p> |
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892 <pb file="add_6787_f034v" o="34v" n="68"/> |
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893 <pb file="add_6787_f035" o="35" n="69"/> |
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894 <div xml:id="echoid-div14" type="page_commentary" level="2" n="14"> |
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895 <p> |
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896 <s xml:id="echoid-s74" xml:space="preserve">[<emph style="it">Note: |
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897 <p> |
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898 <s xml:id="echoid-s74" xml:space="preserve"> |
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899 Further work on spherical triangles. |
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900 </s> |
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901 </p> |
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902 </emph>] |
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903 <lb/><lb/></s></p></div> |
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904 <head xml:id="echoid-head14" xml:space="preserve" xml:lang="lat"> |
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905 (10. |
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906 </head> |
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907 <p xml:lang="lat"> |
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908 <s xml:id="echoid-s76" xml:space="preserve"> |
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909 Duo triangula <lb/> |
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910 sub canone 1<emph style="super">æ</emph> analogiæ <lb/> |
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911 et duobus primis <lb/> |
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912 casibus. |
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913 <lb/>[<emph style="it">tr: |
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914 Two triangles under the canon of the first ratio and the two first cases. |
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915 </emph>]<lb/> |
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916 </s> |
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917 </p> |
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918 <p xml:lang="lat"> |
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919 <s xml:id="echoid-s77" xml:space="preserve"> |
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920 (operationum designationes <lb/> |
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921 sunt in Chartis nostris de <lb/> |
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922 triangulis sphæricis.) |
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923 <lb/>[<emph style="it">tr: |
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924 (the specifications of the operations are in my sheets on spherical triangles) |
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925 </emph>]<lb/> |
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926 </s> |
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927 </p> |
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928 <pb file="add_6787_f035v" o="35v" n="70"/> |
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929 <pb file="add_6787_f036" o="36" n="71"/> |
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930 <div xml:id="echoid-div15" type="page_commentary" level="2" n="15"> |
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931 <p> |
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932 <s xml:id="echoid-s78" xml:space="preserve">[<emph style="it">Note: |
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933 <p> |
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934 <s xml:id="echoid-s78" xml:space="preserve"> |
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935 Further work on spherical triangles. |
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936 </s> |
|
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|
937 </p> |
|
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|
938 </emph>] |
|
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|
939 <lb/><lb/></s></p></div> |
|
22d6a63640c6
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|
940 <head xml:id="echoid-head15" xml:space="preserve" xml:lang="lat"> |
|
22d6a63640c6
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|
941 (11. |
|
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|
942 </head> |
|
22d6a63640c6
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|
943 <p xml:lang="lat"> |
|
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|
944 <s xml:id="echoid-s80" xml:space="preserve"> |
|
22d6a63640c6
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|
945 Analogia per sinus versos |
|
22d6a63640c6
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|
946 <lb/>[<emph style="it">tr: |
|
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|
947 Ratio in terms of versed sines. |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
948 </emph>]<lb/> |
|
22d6a63640c6
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|
949 </s> |
|
22d6a63640c6
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|
950 </p> |
|
22d6a63640c6
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|
951 <pb file="add_6787_f036v" o="36v" n="72"/> |
|
22d6a63640c6
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|
952 <pb file="add_6787_f037" o="37" n="73"/> |
|
22d6a63640c6
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|
953 <div xml:id="echoid-div16" type="page_commentary" level="2" n="16"> |
|
22d6a63640c6
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|
954 <p> |
|
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|
955 <s xml:id="echoid-s81" xml:space="preserve">[<emph style="it">Note: |
|
22d6a63640c6
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|
956 <p> |
|
22d6a63640c6
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|
957 <s xml:id="echoid-s81" xml:space="preserve"> |
|
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|
958 Further work on spherical triangles. |
|
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|
959 </s> |
|
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|
960 </p> |
|
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|
961 </emph>] |
|
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|
962 <lb/><lb/></s></p></div> |
|
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|
963 <head xml:id="echoid-head16" xml:space="preserve" xml:lang="lat"> |
|
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|
964 (12. |
|
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|
965 </head> |
|
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|
966 <p xml:lang="lat"> |
|
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|
967 <s xml:id="echoid-s83" xml:space="preserve"> |
|
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|
968 Analogia per sinus versos |
|
22d6a63640c6
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|
969 <lb/>[<emph style="it">tr: |
|
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|
970 Ratio in terms of versed sines. |
|
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|
971 </emph>]<lb/> |
|
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|
972 </s> |
|
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|
973 </p> |
|
22d6a63640c6
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|
974 <pb file="add_6787_f037v" o="37v" n="74"/> |
|
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|
975 <pb file="add_6787_f038" o="38" n="75"/> |
|
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|
976 <div xml:id="echoid-div17" type="page_commentary" level="2" n="17"> |
|
22d6a63640c6
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|
977 <p> |
|
22d6a63640c6
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|
978 <s xml:id="echoid-s84" xml:space="preserve">[<emph style="it">Note: |
|
22d6a63640c6
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|
979 <p> |
|
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|
980 <s xml:id="echoid-s84" xml:space="preserve"> |
|
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|
981 Further work on spherical triangles. |
|
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|
982 </s> |
|
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|
983 </p> |
|
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|
984 </emph>] |
|
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|
985 <lb/><lb/></s></p></div> |
|
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|
986 <head xml:id="echoid-head17" xml:space="preserve" xml:lang="lat"> |
|
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|
987 (13. |
|
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|
988 </head> |
|
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|
989 <p xml:lang="lat"> |
|
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|
990 <s xml:id="echoid-s86" xml:space="preserve"> |
|
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|
991 Analogia per sinus versos |
|
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|
992 <lb/>[<emph style="it">tr: |
|
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|
993 Ratio in terms of versed sines. |
|
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|
994 </emph>]<lb/> |
|
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|
995 </s> |
|
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|
996 </p> |
|
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|
997 <pb file="add_6787_f038v" o="38v" n="76"/> |
|
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|
998 <pb file="add_6787_f039" o="39" n="77"/> |
|
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|
999 <div xml:id="echoid-div18" type="page_commentary" level="2" n="18"> |
|
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|
1000 <p> |
|
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|
1001 <s xml:id="echoid-s87" xml:space="preserve">[<emph style="it">Note: |
|
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|
1002 <p> |
|
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|
1003 <s xml:id="echoid-s87" xml:space="preserve"> |
|
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|
1004 Further work on spherical triangles. |
|
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|
1005 </s> |
|
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|
1006 </p> |
|
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|
1007 </emph>] |
|
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|
1008 <lb/><lb/></s></p></div> |
|
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|
1009 <head xml:id="echoid-head18" xml:space="preserve" xml:lang="lat"> |
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|
1010 (14. |
|
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|
1011 </head> |
|
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1012 <p xml:lang="lat"> |
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1013 <s xml:id="echoid-s89" xml:space="preserve"> |
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|
1014 Analogia per sinus versos |
|
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1015 <lb/>[<emph style="it">tr: |
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|
1016 Ratio in terms of versed sines. |
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1017 </emph>]<lb/> |
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1018 </s> |
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1019 </p> |
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1020 <pb file="add_6787_f039v" o="39v" n="78"/> |
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1021 <pb file="add_6787_f040" o="40" n="79"/> |
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1022 <div xml:id="echoid-div19" type="page_commentary" level="2" n="19"> |
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|
1023 <p> |
|
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1024 <s xml:id="echoid-s90" xml:space="preserve">[<emph style="it">Note: |
|
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1025 <p> |
|
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1026 <s xml:id="echoid-s90" xml:space="preserve"> |
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1027 Further work on spherical triangles. |
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1028 </s> |
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1029 </p> |
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1030 </emph>] |
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1031 <lb/><lb/></s></p></div> |
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1032 <head xml:id="echoid-head19" xml:space="preserve" xml:lang="lat"> |
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1033 (15. <lb/> |
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1034 De 5<emph style="super">o</emph>, et 6<emph style="super">o</emph>, casu |
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1035 <foreign xml:lang="gre">ton syntomon</foreign> |
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1036 <lb/>[<emph style="it">tr: |
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1037 On the 5th and 6th cases of syntomon. |
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1038 </emph>]<lb/> |
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1039 </head> |
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1040 <p xml:lang="lat"> |
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1041 <s xml:id="echoid-s92" xml:space="preserve"> |
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1042 5<emph style="super">us</emph> casus est, quando uterque arcus sit maior quadrante <lb/> |
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1043 et aggregatum maius 3<emph style="super">bus</emph> quadrantibus. |
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1044 <lb/>[<emph style="it">tr: |
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1045 The 5th case is when either arc is greater than a quadrant, and the sum is greater than three quadrants. |
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1046 </emph>]<lb/> |
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1047 </s> |
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1048 </p> |
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1049 <p xml:lang="lat"> |
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1050 <s xml:id="echoid-s93" xml:space="preserve"> |
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1051 Dico etiam quod: in utroque casu: <lb/> |
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1052 <lb/>[...]<lb/> <lb/> |
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1053 Quod fuit demosntradum. |
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1054 <lb/>[<emph style="it">tr: |
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1055 I also say that in eithe case <lb/> |
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1056 <lb/>[...]<lb/> <lb/> |
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1057 Which was to be demonstrated. |
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1058 </emph>]<lb/> |
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1059 </s> |
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1060 </p> |
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1061 <p xml:lang="lat"> |
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1062 <s xml:id="echoid-s94" xml:space="preserve"> |
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1063 6<emph style="super">us</emph> casus est, quando uterque arcus sit maior quadrante <lb/> |
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1064 et aggregatum minus 3<emph style="super">bus</emph> quadrantibus. |
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1065 <lb/>[<emph style="it">tr: |
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1066 The 6th case is when either arc is greater than a quadrant, and the sum is less than three quadrants. |
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1067 </emph>]<lb/> |
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1068 </s> |
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1069 </p> |
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1070 <pb file="add_6787_f040v" o="40v" n="80"/> |
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1071 <pb file="add_6787_f041" o="41" n="81"/> |
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1072 <div xml:id="echoid-div20" type="page_commentary" level="2" n="20"> |
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1073 <p> |
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1074 <s xml:id="echoid-s95" xml:space="preserve">[<emph style="it">Note: |
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1075 <p> |
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1076 <s xml:id="echoid-s95" xml:space="preserve"> |
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1077 Further work on spherical triangles. |
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1078 </s> |
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1079 </p> |
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1080 </emph>] |
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1081 <lb/><lb/></s></p></div> |
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1082 <head xml:id="echoid-head20" xml:space="preserve" xml:lang="lat"> |
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1083 (16. <lb/> |
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1084 <foreign xml:lang="gre">ton syntomon</foreign> ruminatio. 6 casus exhibantur et omnes <lb/> |
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1085 varia enunicantur et designantur; ut magis canonicæ <lb/> |
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1086 formæ ad usus seligantur. |
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1087 <lb/>[<emph style="it">tr: |
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1088 Further thoughts on syntomon. <lb/> |
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1089 6 cases are exhibeted and all varations enunicated and specified; |
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1090 so that canonical forms can further be picked out in practice. |
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1091 </emph>]<lb/> |
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1092 </head> |
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1093 <pb file="add_6787_f041v" o="41v" n="82"/> |
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1094 <pb file="add_6787_f042" o="42" n="83"/> |
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1095 <div xml:id="echoid-div21" type="page_commentary" level="2" n="21"> |
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1096 <p> |
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1097 <s xml:id="echoid-s97" xml:space="preserve">[<emph style="it">Note: |
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1098 <p> |
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1099 <s xml:id="echoid-s97" xml:space="preserve"> |
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1100 Further work on spherical triangles. |
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1101 </s> |
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1102 </p> |
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1103 </emph>] |
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1104 <lb/><lb/></s></p></div> |
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1105 <head xml:id="echoid-head21" xml:space="preserve" xml:lang="lat"> |
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1106 (17. <lb/> |
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1107 sex casuum <foreign xml:lang="gre">ton syntomon</foreign> <lb/> |
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1108 duæ selectæ formæ. |
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1109 <lb/>[<emph style="it">tr: |
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1110 Six cases of Syntomon; <lb/> |
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1111 two chosen forms. |
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1112 </emph>]<lb/> |
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1113 </head> |
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1114 <p xml:lang="lat"> |
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1115 <s xml:id="echoid-s99" xml:space="preserve"> |
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1116 Exempla analogiarum, accommodata casibus. |
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1117 <lb/>[<emph style="it">tr: |
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1118 Examples of ratios, accommodated to cases. |
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1119 </emph>]<lb/> |
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1120 </s> |
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1121 </p> |
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1122 <p xml:lang="lat"> |
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1123 <s xml:id="echoid-s100" xml:space="preserve"> |
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1124 sex sunt diversæ analogiæ sub speciebus arcuum <lb/> |
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1125 cum tamen duæ solummodo sub numerus sinuum. <lb/> |
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1126 Unde manifestum quod 1<emph style="super">us</emph> et 2<emph style="super">us</emph> casus |
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1127 <foreign xml:lang="gre">ton syntomon</foreign> possunt <lb/> |
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1128 satis ad operationes. reliquæ tamen ad argumentationes sunt <lb/> |
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1129 utilis. |
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1130 <lb/>[<emph style="it">tr: |
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1131 There are six different ratios for types of arc, but with only two for sines in numbers. <lb/> |
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1132 From which is is clear that the 1st and 2nd cases may be enough for working; |
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1133 nevertheless, the others are useful for arguments. |
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1134 </emph>]<lb/> |
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1135 </s> |
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1136 </p> |
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1137 <pb file="add_6787_f042v" o="42v" n="84"/> |
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1138 <pb file="add_6787_f043" o="43" n="85"/> |
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1139 <div xml:id="echoid-div22" type="page_commentary" level="2" n="22"> |
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1140 <p> |
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1141 <s xml:id="echoid-s101" xml:space="preserve">[<emph style="it">Note: |
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1142 <p> |
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1143 <s xml:id="echoid-s101" xml:space="preserve"> |
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1144 Further work on spherical triangles. |
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1145 </s> |
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1146 </p> |
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1147 </emph>] |
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1148 <lb/><lb/></s></p></div> |
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1149 <head xml:id="echoid-head22" xml:space="preserve" xml:lang="lat"> |
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1150 (18. <lb/> |
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1151 Exempla operandi per <foreign xml:lang="gre">syntomon</foreign>, <lb/> |
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1152 secundum duas formas selectas. |
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1153 <lb/>[<emph style="it">tr: |
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1154 Examples of working with Syntomon according to two chosen forms. |
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1155 </emph>]<lb/> |
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1156 </head> |
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1157 <pb file="add_6787_f043v" o="43v" n="86"/> |
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1158 <pb file="add_6787_f044" o="44" n="87"/> |
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1159 <pb file="add_6787_f044v" o="44v" n="88"/> |
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1160 <pb file="add_6787_f045" o="45" n="89"/> |
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1161 <pb file="add_6787_f045v" o="45v" n="90"/> |
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1162 <pb file="add_6787_f046" o="46" n="91"/> |
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1163 <pb file="add_6787_f046v" o="46v" n="92"/> |
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1164 <pb file="add_6787_f047" o="47" n="93"/> |
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1165 <pb file="add_6787_f047v" o="47v" n="94"/> |
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1166 <pb file="add_6787_f048" o="48" n="95"/> |
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1167 <pb file="add_6787_f048v" o="48v" n="96"/> |
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1168 <pb file="add_6787_f049" o="49" n="97"/> |
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1169 <pb file="add_6787_f049v" o="49v" n="98"/> |
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1170 <pb file="add_6787_f050" o="50" n="99"/> |
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1171 <pb file="add_6787_f050v" o="50v" n="100"/> |
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1172 <pb file="add_6787_f051" o="51" n="101"/> |
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1173 <div xml:id="echoid-div23" type="page_commentary" level="2" n="23"> |
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1174 <p> |
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1175 <s xml:id="echoid-s103" xml:space="preserve">[<emph style="it">Note: |
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1176 <p> |
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1177 <s xml:id="echoid-s103" xml:space="preserve"> |
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1178 This page contains references to both Regiomontanus and Clavius, |
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1179 who both gave a version of this theorem. <lb/> |
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1180 The reference to Regiomontanus is to his <emph style="it">De triangulis omnimodis libri quinque</emph> (1533), |
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1181 Book V, Proposition II. (For another reference to the same proposition, see also Add MS 6782, f. 490.) |
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1182 </s> |
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1183 <lb/> |
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1184 <quote xml:lang="lat"> |
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1185 V.II In omni triangulo sphaerali ex arcubus circulorum magnorum constante, |
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1186 proportio sinus uersi anguli cuislibet ad differentiam duorum sinuum uersorum, |
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1187 quorum unus est lateris eum angulum subtendentis: |
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1188 alius uerò differentiae duorum arcuum ipsi angulo circumiacentium |
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1189 est tanquam proportio quadrati sinus recti totius ad id, |
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1190 quod sub sinibus arcuum dicto angulo circumpositorum continetur rectangulum |
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1191 </quote> |
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1192 <lb/> |
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1193 <quote> |
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1194 In all spherical triangles composed from great arcs of circles, the ratio of the versed sine of any angle |
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1195 to the difference of two versed sines, one of which is the side subtending the angle, |
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1196 the other the difference of the two arcs adjacent to the angle, |
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1197 is the proportion of the the square of the whole sine |
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1198 to the product of the sines of the surrounding arcs by which the said angle is contained. |
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1199 </quote> |
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1200 <lb/> |
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1201 <s xml:id="echoid-s104" xml:space="preserve"> |
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1202 The reference to Clavius is to his <emph style="it">Triangula sphærica</emph> in |
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1203 <emph style="it">Triangula rectilinea, atque sphaerica</emph> (1586). |
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1204 Proposition 58, on page 445. |
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1205 </s> |
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1206 <lb/> |
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1207 <quote xml:lang="lat"> |
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1208 Theorema 56, Propositio 58. <lb/> |
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1209 In omni triangulo sphærico, cuius duo arcus sint inæquales; |
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1210 quadratum sinus totius ad rectangulum sub sinubus rectis duorum arcuum inæqulium contentum, |
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1211 eandem proportionem habet, quam sinus versus anguli a dictis arcubus comprehensi |
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1212 ad differentiam duorum sinuum versorum, quorum vnus differentiæ eorundem arcuum debetur, |
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1213 alter vero tertio arcui, qui prædicto angulo oppostitus est, respondet. |
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1214 </quote> |
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1215 <lb/> |
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1216 <quote> |
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1217 In all spherical triangles, whose two arcs are unequal, |
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1218 the square of the whole sine to the product of the sines of the two unequal arcs |
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1219 is in the same ratio as the versed sine of the angle between the said arcs |
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1220 to the difference of two versed sines, one of which is of the difference of the arcs, |
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1221 the other corresponding to the third arc, which is opposite the aforesaid angle. |
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1222 </quote> |
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1223 <lb/> |
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1224 <s xml:id="echoid-s105" xml:space="preserve"> |
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1225 Harrot translates Clavius's statement into symbols for the particular triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>a</mi><mi>b</mi></mstyle></math> shown in his diagram. |
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1226 He then goes on to prove that the versed sine of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi><mo>-</mo><mi>d</mi><mi>b</mi></mstyle></math> is greater than the versed sine of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. <lb/> |
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1227 For another version of this page, see Add MS 6787, f. 27. |
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1228 </s> |
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1229 </p> |
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1230 </emph>] |
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1231 <lb/><lb/></s></p></div> |
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1232 <head xml:id="echoid-head23" xml:space="preserve" xml:lang="lat"> |
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1233 Analogia per sinus versos, et universalis ad triangula <lb/> |
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1234 sphærica cuiuscunque <emph style="st">affectionis</emph> conditionis. |
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1235 <lb/>[<emph style="it">tr: |
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1236 Ratio by versed sines, and generally for a spherical trianlge under any conditions. |
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1237 </emph>]<lb/> |
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1238 </head> |
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1239 <p xml:lang="lat"> |
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1240 <s xml:id="echoid-s107" xml:space="preserve"> |
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1241 Demonstratur a Regiomontano <lb/> |
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1242 lib. 5<emph style="super">to</emph>. pr. 2, de triang. <lb/> |
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1243 a clavio pr. 58. de sphær. Tri. <lb/> |
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1244 et ab alijs Trigonistis. |
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1245 <lb/>[<emph style="it">tr: |
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1246 Demonstrated by Regiomontanus in <emph style="it">De triangulis</emph>, Book 5, Proposition 2. <lb/> |
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1247 by Clavius in Proposition 58 of <emph style="it">De sphæricis triangulis</emph> <lb/> |
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1248 and by other triangulists. |
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1249 </emph>]<lb/> |
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1250 </s> |
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1251 </p> |
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1252 <p xml:lang="lat"> |
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1253 <s xml:id="echoid-s108" xml:space="preserve"> |
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1254 Dico quod: <lb/>[...]<lb/> (universitaliter) |
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1255 <lb/>[<emph style="it">tr: |
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1256 I say that <lb/>[...]<lb/> (generally) |
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1257 </emph>]<lb/> |
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1258 </s> |
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1259 </p> |
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1260 <p xml:lang="lat"> |
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1261 <s xml:id="echoid-s109" xml:space="preserve"> |
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1262 Consectarium |
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1263 <lb/>[<emph style="it">tr: |
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1264 Consequence |
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1265 </emph>]<lb/> |
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1266 </s> |
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1267 <lb/> |
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1268 <s xml:id="echoid-s110" xml:space="preserve"> |
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1269 ponatur quod: <lb/> |
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1270 In triangulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi><mi>b</mi></mstyle></math>, Datis duobus lateribus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math>; cum ang: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> <lb/> |
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1271 queratur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. <lb/> |
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1272 per superiorem analogiam, sit quarta proportionalis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. <lb/> |
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1273 <lb/>[...]<lb/> <lb/> |
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1274 datur igitur, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. |
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1275 <lb/>[<emph style="it">tr: |
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1276 It is supposed that, in triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi><mi>b</mi></mstyle></math>, from given two sides <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math> with the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, |
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1277 there is sought <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>. <lb/> |
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1278 by the above ratio, let the given fourth proportional be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math> <lb/> |
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1279 <omissions/> <lb/> |
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1280 therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> is given |
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1281 </emph>]<lb/> |
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1282 </s> |
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1283 </p> |
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1284 <p xml:lang="lat"> |
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1285 <s xml:id="echoid-s111" xml:space="preserve"> |
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1286 <emph style="st">omittuntur sequentia</emph> |
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1287 <lb/>[<emph style="it">tr: |
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1288 The following are omitted. |
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1289 </emph>]<lb/> |
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1290 </s> |
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1291 </p> |
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1292 <pb file="add_6787_f051v" o="51v" n="102"/> |
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1293 <pb file="add_6787_f052" o="52" n="103"/> |
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1294 <pb file="add_6787_f052v" o="52v" n="104"/> |
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1295 <pb file="add_6787_f053" o="53" n="105"/> |
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1296 <div xml:id="echoid-div24" type="page_commentary" level="2" n="24"> |
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1297 <p> |
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1298 <s xml:id="echoid-s112" xml:space="preserve">[<emph style="it">Note: |
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1299 <p> |
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1300 <s xml:id="echoid-s112" xml:space="preserve"> |
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1301 Another version of the information in Add MS 6787, f. 54. |
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1302 </s> |
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1303 </p> |
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1304 </emph>] |
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1305 <lb/><lb/></s></p></div> |
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1306 <head xml:id="echoid-head24" xml:space="preserve" xml:lang="lat"> |
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1307 5) Ad calculum sinuum trium differentiarum succedentium |
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1308 <lb/>[<emph style="it">tr: |
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1309 For the calculation of three successive differences of sines. |
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1310 </emph>]<lb/> |
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1311 </head> |
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1312 <p xml:lang="lat"> |
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1313 <s xml:id="echoid-s114" xml:space="preserve"> |
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1314 Universalis aequatio. examinata <lb/> |
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1315 per rationales progressiones, charta 6a <lb/> |
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1316 et per sinuus. |
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1317 <lb/>[<emph style="it">tr: |
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1318 General equation, examined by rational progressions, sheet 6a, and by sines. |
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1319 </emph>]<lb/> |
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1320 </s> |
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1321 </p> |
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1322 <p xml:lang="lat"> |
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1323 <s xml:id="echoid-s115" xml:space="preserve"> |
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1324 Ista aequatione fit ex reductione <lb/> |
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1325 superioris. |
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1326 <lb/>[<emph style="it">tr: |
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1327 This equation arises by reduction of the above. |
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1328 </emph>]<lb/> |
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1329 </s> |
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1330 </p> |
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1331 <pb file="add_6787_f053v" o="53v" n="106"/> |
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1332 <pb file="add_6787_f054" o="54" n="107"/> |
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1333 <div xml:id="echoid-div25" type="page_commentary" level="2" n="25"> |
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1334 <p> |
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1335 <s xml:id="echoid-s116" xml:space="preserve">[<emph style="it">Note: |
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1336 <p> |
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1337 <s xml:id="echoid-s116" xml:space="preserve"> |
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1338 The columns show the first, second and third differences of an interpolated table |
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1339 with constant third difference. |
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1340 </s> |
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1341 </p> |
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1342 </emph>] |
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1343 <lb/><lb/></s></p></div> |
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1344 <head xml:id="echoid-head25" xml:space="preserve"> |
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1345 4) |
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1346 </head> |
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1347 <pb file="add_6787_f054v" o="54v" n="108"/> |
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1348 <pb file="add_6787_f055" o="55" n="109"/> |
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1349 <div xml:id="echoid-div26" type="page_commentary" level="2" n="26"> |
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1350 <p> |
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1351 <s xml:id="echoid-s118" xml:space="preserve">[<emph style="it">Note: |
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1352 <p> |
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1353 <s xml:id="echoid-s118" xml:space="preserve"> |
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1354 Formulae for the entries in a table that has been interpolated to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> times its original length; |
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1355 for full details see the 'Magisteria magna', Add MS 6782, f. 107 to f. 146v. <lb/> |
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1356 The third difference is constant, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>n</mi><mi>n</mi><mi>n</mi></mstyle></math>. <lb/> |
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1357 The second differences begin with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, the first differences begin with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>2</mn></msup></mrow></mstyle></math>; |
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1358 these are superscripts, not powers. <lb/> |
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1359 The differences alternately increase and decrease, as in a table of sines. <lb/> |
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1360 Values are calculated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>th, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>n</mi></mstyle></math>th, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>n</mi></mstyle></math>th entries |
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1361 using the result obtained on Add MS 6787. f. 56. |
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1362 </s> |
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1363 </p> |
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1364 </emph>] |
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1365 <lb/><lb/></s></p></div> |
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1366 <head xml:id="echoid-head26" xml:space="preserve"> |
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1367 3) |
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1368 </head> |
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1369 <pb file="add_6787_f055v" o="55v" n="110"/> |
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1370 <pb file="add_6787_f056" o="56" n="111"/> |
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1371 <div xml:id="echoid-div27" type="page_commentary" level="2" n="27"> |
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1372 <p> |
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1373 <s xml:id="echoid-s120" xml:space="preserve">[<emph style="it">Note: |
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1374 <p> |
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1375 <s xml:id="echoid-s120" xml:space="preserve"> |
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1376 Calculations of formulae for the rows of Pascal's triangle, beginning from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>n</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>n</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mi>n</mi></mstyle></math>. <lb/> |
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1377 In each case the numerators are denoted <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>1</mn></msup></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>2</mn></msup></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>3</mn></msup></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>4</mn></msup></mrow></mstyle></math>; |
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1378 these are superscripts, not powers. <lb/> |
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1379 For calculations of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>5</mn></msup></mrow></mstyle></math> see Add MS 6787, f. 252v. |
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1380 </s> |
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1381 </p> |
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1382 </emph>] |
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1383 <lb/><lb/></s></p></div> |
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1384 <head xml:id="echoid-head27" xml:space="preserve"> |
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1385 2) |
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1386 </head> |
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1387 <pb file="add_6787_f056v" o="56v" n="112"/> |
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1388 <pb file="add_6787_f057v" o="57v" n="113"/> |
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1389 <pb file="add_6787_f058" o="58" n="114"/> |
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1390 <div xml:id="echoid-div28" type="page_commentary" level="2" n="28"> |
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1391 <p> |
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1392 <s xml:id="echoid-s122" xml:space="preserve">[<emph style="it">Note: |
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1393 <p> |
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1394 <s xml:id="echoid-s122" xml:space="preserve"> |
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1395 On the left is a table of 'pretend' sines with constant third difference 81. <lb/> |
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1396 The final row of the tables is 7240, 600, 405, 81. <lb/> |
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1397 The task is to interpolate the table with two new entries between each existing entry, |
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1398 using the formulae calculated in the previous sheets. <lb/> |
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1399 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>=</mo><mn>3</mn></mstyle></math> by hypothesis. <lb/> |
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1400 Thus the constant difference in the interpolated table will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>8</mn><mn>1</mn></mrow><mrow><mn>2</mn><mn>7</mn></mrow></mfrac><mo>=</mo><mn>3</mn></mstyle></math>. <lb/> |
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1401 Further calculations give <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mo>=</mo><mn>4</mn><mn>5</mn></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>2</mn></msup></mrow></mstyle></math> for the first new entries |
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1402 after the row beginning 6640. |
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1403 </s> |
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1404 </p> |
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1405 </emph>] |
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1406 <lb/><lb/></s></p></div> |
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1407 <head xml:id="echoid-head28" xml:space="preserve"> |
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1408 6) |
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1409 </head> |
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1410 <p xml:lang="lat"> |
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1411 <s xml:id="echoid-s124" xml:space="preserve"> |
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1412 data |
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1413 <lb/>[<emph style="it">tr: |
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1414 given |
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1415 </emph>]<lb/> |
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1416 </s> |
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1417 </p> |
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1418 <p xml:lang="lat"> |
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1419 <s xml:id="echoid-s125" xml:space="preserve"> |
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1420 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>=</mo><mn>3</mn></mstyle></math> per hypothesi numero dividendum <lb/> |
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1421 partium. |
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1422 <lb/>[<emph style="it">tr: |
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1423 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>=</mo><mn>3</mn></mstyle></math> by hypothesis, the number of parts to be divided. |
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1424 </emph>]<lb/> |
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1425 </s> |
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1426 </p> |
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1427 <pb file="add_6787_f058v" o="58v" n="115"/> |
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1428 <pb file="add_6787_f059" o="59" n="116"/> |
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1429 <div xml:id="echoid-div29" type="page_commentary" level="2" n="29"> |
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1430 <p> |
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1431 <s xml:id="echoid-s126" xml:space="preserve">[<emph style="it">Note: |
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1432 <p> |
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1433 <s xml:id="echoid-s126" xml:space="preserve"> |
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1434 At the top of the page is a table of sines for every 45 minutes, with fourth differences. |
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1435 The sine of 48.45' is highlighted. <lb/> |
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1436 In the centre is a table of sines for every 15 minutes, with third differences. |
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1437 The sine of 48.45' is in the final row. <lb/> |
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1438 At the bottom is a table of sines for every 15 minutes, calculated from the row for 48.45 |
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1439 on the assumption that the fourth difference is constant at |
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1440 56 for the upper half of the table, 55 for the lower half. |
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1441 </s> |
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1442 </p> |
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1443 </emph>] |
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1444 <lb/><lb/></s></p></div> |
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1445 <head xml:id="echoid-head29" xml:space="preserve" xml:lang="lat"> |
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1446 Ad calculum sinuum, per progressiones. |
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1447 <lb/>[<emph style="it">tr: |
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1448 For the calculation of sines, by progressisons |
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1449 </emph>]<lb/> |
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1450 </head> |
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1451 <p xml:lang="lat"> |
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1452 <s xml:id="echoid-s128" xml:space="preserve"> |
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|
1453 data |
|
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1454 <lb/>[<emph style="it">tr: |
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|
1455 given |
|
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1456 </emph>]<lb/> |
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1457 </s> |
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1458 </p> |
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1459 <p xml:lang="lat"> |
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1460 <s xml:id="echoid-s129" xml:space="preserve"> |
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|
1461 E canone sinuum |
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1462 <lb/>[<emph style="it">tr: |
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|
1463 From the table of sines |
|
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|
1464 </emph>]<lb/> |
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1465 </s> |
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|
1466 </p> |
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1467 <p xml:lang="lat"> |
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1468 <s xml:id="echoid-s130" xml:space="preserve"> |
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|
1469 Per caclulum |
|
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1470 <lb/>[<emph style="it">tr: |
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|
1471 By calculation |
|
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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_6787_f059v" o="59v" n="117"/> |
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|
1476 <pb file="add_6787_f060" o="60" n="118"/> |
|
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|
1477 <pb file="add_6787_f060v" o="60v" n="119"/> |
|
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|
1478 <pb file="add_6787_f061" o="61" n="120"/> |
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|
1479 <pb file="add_6787_f061v" o="61v" n="121"/> |
|
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|
1480 <pb file="add_6787_f062" o="62" n="122"/> |
|
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|
1481 <pb file="add_6787_f062v" o="62v" n="123"/> |
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|
1482 <pb file="add_6787_f063" o="63" n="124"/> |
|
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|
1483 <div xml:id="echoid-div30" type="page_commentary" level="2" n="30"> |
|
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|
1484 <p> |
|
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|
1485 <s xml:id="echoid-s131" xml:space="preserve">[<emph style="it">Note: |
|
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|
1486 <p> |
|
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|
1487 <s xml:id="echoid-s131" xml:space="preserve"> |
|
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|
1488 Selections of sines from the beginning, end, and middle of the table, |
|
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|
1489 each showing a pattern of alternately decreasing and increasing columns of differences. <lb/> |
|
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|
1490 </s> |
|
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|
1491 </p> |
|
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|
1492 </emph>] |
|
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|
1493 <lb/><lb/></s></p></div> |
|
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|
1494 <p xml:lang="lat"> |
|
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|
1495 <s xml:id="echoid-s133" xml:space="preserve"> |
|
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|
1496 <reg norm="Differentiae" type="abbr">Dræ</reg> |
|
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|
1497 <lb/>[<emph style="it">tr: |
|
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|
1498 Differences |
|
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|
1499 </emph>]<lb/> |
|
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|
1500 </s> |
|
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|
1501 </p> |
|
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|
1502 <p xml:lang="lat"> |
|
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|
1503 <s xml:id="echoid-s134" xml:space="preserve"> |
|
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|
1504 In principio tablum sinuum |
|
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|
1505 <lb/>[<emph style="it">tr: |
|
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|
1506 From the beginning of the table of sines |
|
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|
1507 </emph>]<lb/> |
|
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|
1508 </s> |
|
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|
1509 </p> |
|
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|
1510 <p xml:lang="lat"> |
|
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|
1511 <s xml:id="echoid-s135" xml:space="preserve"> |
|
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|
1512 In finis |
|
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|
1513 <lb/>[<emph style="it">tr: |
|
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|
1514 From the end |
|
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|
1515 </emph>]<lb/> |
|
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|
1516 </s> |
|
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|
1517 </p> |
|
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|
1518 <p xml:lang="lat"> |
|
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|
1519 <s xml:id="echoid-s136" xml:space="preserve"> |
|
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|
1520 In medio |
|
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|
1521 <lb/>[<emph style="it">tr: |
|
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|
1522 From the middle |
|
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|
1523 </emph>]<lb/> |
|
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|
1524 </s> |
|
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|
1525 </p> |
|
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|
1526 <p xml:lang="lat"> |
|
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|
1527 <s xml:id="echoid-s137" xml:space="preserve"> |
|
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|
1528 In principio iterum |
|
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|
1529 <lb/>[<emph style="it">tr: |
|
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|
1530 From the beginning again |
|
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|
1531 </emph>]<lb/> |
|
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|
1532 </s> |
|
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|
1533 </p> |
|
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|
1534 <pb file="add_6787_f063v" o="63v" n="125"/> |
|
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|
1535 <pb file="add_6787_f064" o="64" n="126"/> |
|
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1536 <pb file="add_6787_f064v" o="64v" n="127"/> |
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1537 <pb file="add_6787_f065" o="65" n="128"/> |
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1538 <pb file="add_6787_f065v" o="65v" n="129"/> |
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1539 <pb file="add_6787_f066" o="66" n="130"/> |
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1540 <div xml:id="echoid-div31" type="page_commentary" level="2" n="31"> |
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1541 <p> |
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1542 <s xml:id="echoid-s138" xml:space="preserve">[<emph style="it">Note: |
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1543 <p> |
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1544 <s xml:id="echoid-s138" xml:space="preserve"> |
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1545 The columns show the first, second, third, fourth, and fifth differences of an interpolated table |
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1546 with constant fifth difference. |
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1547 </s> |
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1548 </p> |
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1549 </emph>] |
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1550 <lb/><lb/></s></p></div> |
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1551 <head xml:id="echoid-head30" xml:space="preserve"> |
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|
1552 6) Ad calculum sinuum 5 differentiarum succedentium |
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1553 <lb/>[<emph style="it">tr: |
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|
1554 For the calculation of 5 successive differences of sines |
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1555 </emph>]<lb/> |
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1556 </head> |
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1557 <p xml:lang="lat"> |
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1558 <s xml:id="echoid-s140" xml:space="preserve"> |
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|
1559 Superioris 5, aequationes probatæ sunt <lb/> |
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1560 per rationales progressiones. |
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1561 <lb/>[<emph style="it">tr: |
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|
1562 The above five equations may be proved from known progressions. |
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1563 </emph>]<lb/> |
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1564 </s> |
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1565 </p> |
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1566 <pb file="add_6787_f066v" o="66v" n="131"/> |
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1567 <pb file="add_6787_f067" o="67" n="132"/> |
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1568 <div xml:id="echoid-div32" type="page_commentary" level="2" n="32"> |
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|
1569 <p> |
|
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1570 <s xml:id="echoid-s141" xml:space="preserve">[<emph style="it">Note: |
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|
1571 <p> |
|
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|
1572 <s xml:id="echoid-s141" xml:space="preserve"> |
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|
1573 Third differences of the entries from Add MS 6787, f. 71. |
|
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|
1574 </s> |
|
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|
1575 </p> |
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|
1576 </emph>] |
|
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|
1577 <lb/><lb/></s></p></div> |
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|
1578 <head xml:id="echoid-head31" xml:space="preserve" xml:lang="lat"> |
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|
1579 3) Tertariæ differentiæ |
|
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|
1580 <lb/>[<emph style="it">tr: |
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|
1581 Of third differences |
|
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|
1582 </emph>]<lb/> |
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|
1583 </head> |
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|
1584 <pb file="add_6787_f067v" o="67v" n="133"/> |
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|
1585 <pb file="add_6787_f068" o="68" n="134"/> |
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|
1586 <div xml:id="echoid-div33" type="page_commentary" level="2" n="33"> |
|
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|
1587 <p> |
|
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|
1588 <s xml:id="echoid-s143" xml:space="preserve">[<emph style="it">Note: |
|
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|
1589 <p> |
|
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|
1590 <s xml:id="echoid-s143" xml:space="preserve"> |
|
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|
1591 Second differences of the entries from Add MS 6787, f. 71. |
|
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|
1592 </s> |
|
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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-head32" xml:space="preserve" xml:lang="lat"> |
|
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|
1597 3) Secundariæ differentiæ |
|
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|
1598 <lb/>[<emph style="it">tr: |
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|
1599 Of second differences |
|
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|
1600 </emph>]<lb/> |
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|
1601 </head> |
|
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|
1602 <pb file="add_6787_f068v" o="68v" n="135"/> |
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|
1603 <pb file="add_6787_f069" o="69" n="136"/> |
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|
1604 <div xml:id="echoid-div34" type="page_commentary" level="2" n="34"> |
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|
1605 <p> |
|
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|
1606 <s xml:id="echoid-s145" xml:space="preserve">[<emph style="it">Note: |
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|
1607 <p> |
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|
1608 <s xml:id="echoid-s145" xml:space="preserve"> |
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|
1609 First differences of the entries from Add MS 6787, f. 71. |
|
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|
1610 </s> |
|
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|
1611 </p> |
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|
1612 </emph>] |
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|
1613 <lb/><lb/></s></p></div> |
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|
1614 <head xml:id="echoid-head33" xml:space="preserve" xml:lang="lat"> |
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|
1615 3) Primariæ differentiæ |
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1616 <lb/>[<emph style="it">tr: |
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|
1617 Of first differences |
|
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1618 </emph>]<lb/> |
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1619 </head> |
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|
1620 <pb file="add_6787_f069v" o="69v" n="137"/> |
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|
1621 <pb file="add_6787_f070" o="70" n="138"/> |
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|
1622 <div xml:id="echoid-div35" type="page_commentary" level="2" n="35"> |
|
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1623 <p> |
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1624 <s xml:id="echoid-s147" xml:space="preserve">[<emph style="it">Note: |
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1625 <p> |
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1626 <s xml:id="echoid-s147" xml:space="preserve"> |
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1627 The entries from Add MS 6787, f. 71, now written over the common denominator 120. |
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1628 </s> |
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1629 </p> |
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1630 </emph>] |
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1631 <lb/><lb/></s></p></div> |
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1632 <head xml:id="echoid-head34" xml:space="preserve"> |
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1633 2) |
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1634 </head> |
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1635 <pb file="add_6787_f070v" o="70v" n="139"/> |
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1636 <pb file="add_6787_f071" o="71" n="140"/> |
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1637 <div xml:id="echoid-div36" type="page_commentary" level="2" n="36"> |
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1638 <p> |
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1639 <s xml:id="echoid-s149" xml:space="preserve">[<emph style="it">Note: |
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1640 <p> |
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1641 <s xml:id="echoid-s149" xml:space="preserve"> |
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1642 Formulae for the first differences in a table that has been interpolated to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math> times its original length; |
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1643 for full details see the 'Magisteria magna', Add MS 6782, f. 107 to f. 146v. <lb/> |
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1644 The fourth differences begin with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, the third differences begin with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>2</mn></msup></mrow></mstyle></math>, |
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1645 the second differences begin with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>3</mn></msup></mrow></mstyle></math>, the first differences begin with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>p</mi><mn>4</mn></msup></mrow></mstyle></math>; |
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1646 these are superscripts, not powers. <lb/> |
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1647 The difference columns increase and decrease alternately, as in a table of sines. |
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1648 </s> |
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1649 </p> |
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1650 </emph>] |
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1651 <lb/><lb/></s></p></div> |
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1652 <head xml:id="echoid-head35" xml:space="preserve"> |
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1653 1) |
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1654 </head> |
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1655 <pb file="add_6787_f071v" o="71v" n="141"/> |
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1656 <pb file="add_6787_f072" o="72" n="142"/> |
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1657 <div xml:id="echoid-div37" type="page_commentary" level="2" n="37"> |
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1658 <p> |
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1659 <s xml:id="echoid-s151" xml:space="preserve">[<emph style="it">Note: |
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1660 <p> |
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1661 <s xml:id="echoid-s151" xml:space="preserve"> |
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1662 A difference table with constant fifth difference 1, |
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1663 and with alternating increasing and decreasing columns as for sines. |
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1664 Negative entries appear in every column except the last. |
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1665 </s> |
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1666 </p> |
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1667 </emph>] |
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1668 <lb/><lb/></s></p></div> |
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1669 <head xml:id="echoid-head36" xml:space="preserve"> |
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1670 7) |
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1671 </head> |
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1672 <pb file="add_6787_f072v" o="72v" n="143"/> |
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1673 <pb file="add_6787_f073" o="73" n="144"/> |
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1674 <pb file="add_6787_f073v" o="73v" n="145"/> |
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1675 <pb file="add_6787_f074" o="74" n="146"/> |
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1676 <pb file="add_6787_f074v" o="74v" n="147"/> |
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1677 <pb file="add_6787_f075" o="75" n="148"/> |
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1678 <pb file="add_6787_f075v" o="75v" n="149"/> |
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1679 <pb file="add_6787_f076" o="76" n="150"/> |
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1680 <pb file="add_6787_f076v" o="76v" n="151"/> |
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|
1681 <pb file="add_6787_f077" o="77" n="152"/> |
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|
1682 <pb file="add_6787_f077v" o="77v" n="153"/> |
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1683 <pb file="add_6787_f078" o="78" n="154"/> |
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1684 <pb file="add_6787_f078v" o="78v" n="155"/> |
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1685 <pb file="add_6787_f079" o="79" n="156"/> |
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1686 <pb file="add_6787_f079v" o="79v" n="157"/> |
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1687 <pb file="add_6787_f080" o="80" n="158"/> |
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1688 <pb file="add_6787_f080v" o="80v" n="159"/> |
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1689 <pb file="add_6787_f081" o="81" n="160"/> |
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1690 <pb file="add_6787_f081v" o="81v" n="161"/> |
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1691 <pb file="add_6787_f082" o="82" n="162"/> |
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1692 <pb file="add_6787_f082v" o="82v" n="163"/> |
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1693 <pb file="add_6787_f083" o="83" n="164"/> |
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1694 <pb file="add_6787_f083v" o="83v" n="165"/> |
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|
1695 <p> |
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1696 <s xml:id="echoid-s153" xml:space="preserve"> |
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|
1697 Mr Vincent I received your last letter being wihout water by Mr <lb/> |
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|
1698 Fowler the 28th of March last. Other letter I received about 6 weekes <lb/> |
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|
1699 past of Mr Cook. I received other some whether once or twice I can <lb/> |
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1700 not tell about two years since as I gesse or I thinke some what later. <lb/> |
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|
1701 The first letters by reason of some way into the country I lost <lb/> |
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|
1702 the means that brought them convey my answer. |
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1703 </s> |
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|
1704 </p> |
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|
1705 <pb file="add_6787_f084" o="84" n="166"/> |
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|
1706 <pb file="add_6787_f084v" o="84v" n="167"/> |
|
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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
|
1707 <pb file="add_6787_f085" o="85" n="168"/> |
|
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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|
1708 <pb file="add_6787_f085v" o="85v" n="169"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1709 <pb file="add_6787_f086" o="86" n="170"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1710 <pb file="add_6787_f086v" o="86v" n="171"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1711 <pb file="add_6787_f087" o="87" n="172"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1712 <pb file="add_6787_f087v" o="87v" n="173"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1713 <pb file="add_6787_f088" o="88" n="174"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1714 <pb file="add_6787_f088v" o="88v" n="175"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1715 <pb file="add_6787_f089" o="89" n="176"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1716 <pb file="add_6787_f089v" o="89v" n="177"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1717 <pb file="add_6787_f090" o="90" n="178"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1718 <pb file="add_6787_f090v" o="90v" n="179"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1719 <pb file="add_6787_f091" o="91" n="180"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1720 <pb file="add_6787_f091v" o="91v" n="181"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1721 <pb file="add_6787_f092" o="92" n="182"/> |
|
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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|
1722 <pb file="add_6787_f092v" o="92v" n="183"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1723 <pb file="add_6787_f093" o="93" n="184"/> |
|
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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|
1724 <pb file="add_6787_f093v" o="93v" n="185"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1725 <pb file="add_6787_f094" o="94" n="186"/> |
|
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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|
1726 <pb file="add_6787_f094v" o="94v" n="187"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1727 <pb file="add_6787_f095" o="95" n="188"/> |
|
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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|
1728 <pb file="add_6787_f095v" o="95v" n="189"/> |
|
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
|
1729 <pb file="add_6787_f096" o="96" n="190"/> |
|
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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|
1730 <pb file="add_6787_f096v" o="96v" n="191"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1731 <pb file="add_6787_f097" o="97" n="192"/> |
|
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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|
1732 <pb file="add_6787_f097v" o="97v" n="193"/> |
|
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
|
1733 <pb file="add_6787_f098" o="98" n="194"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff
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|
1734 <pb file="add_6787_f098v" o="98v" n="195"/> |
|
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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|
1735 <pb file="add_6787_f099" o="99" n="196"/> |
|
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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|
1736 <pb file="add_6787_f099v" o="99v" n="197"/> |
|
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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|
1737 <pb file="add_6787_f100" o="100" n="198"/> |
|
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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|
1738 <pb file="add_6787_f100v" o="100v" n="199"/> |
|
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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|
1739 <pb file="add_6787_f101" o="101" n="200"/> |
|
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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|
1740 <pb file="add_6787_f101v" o="101v" n="201"/> |
|
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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|
1741 <pb file="add_6787_f102" o="102" n="202"/> |
|
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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|
1742 <pb file="add_6787_f102v" o="102v" n="203"/> |
|
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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|
1743 <pb file="add_6787_f103" o="103" n="204"/> |
|
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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|
1744 <pb file="add_6787_f103v" o="103v" n="205"/> |
|
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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|
1745 <pb file="add_6787_f104" o="104" n="206"/> |
|
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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|
1746 <pb file="add_6787_f104v" o="104v" n="207"/> |
|
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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|
1747 <pb file="add_6787_f105" o="105" n="208"/> |
|
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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|
1748 <pb file="add_6787_f105v" o="105v" n="209"/> |
|
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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|
1749 <pb file="add_6787_f106" o="106" n="210"/> |
|
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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|
1750 <pb file="add_6787_f106v" o="106v" n="211"/> |
|
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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|
1751 <pb file="add_6787_f107" o="107" n="212"/> |
|
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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|
1752 <pb file="add_6787_f107v" o="107v" n="213"/> |
|
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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|
1753 <pb file="add_6787_f108" o="108" n="214"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff
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|
1754 <pb file="add_6787_f108v" o="108v" n="215"/> |
|
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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|
1755 <pb file="add_6787_f109" o="109" n="216"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff
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|
1756 <pb file="add_6787_f109v" o="109v" n="217"/> |
|
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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|
1757 <pb file="add_6787_f110" o="110" n="218"/> |
|
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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|
1758 <pb file="add_6787_f110v" o="110v" n="219"/> |
|
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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|
1759 <pb file="add_6787_f111" o="111" n="220"/> |
|
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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|
1760 <pb file="add_6787_f111v" o="111v" n="221"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1761 <div xml:id="echoid-div38" type="page_commentary" level="2" n="38"> |
|
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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|
1762 <p> |
|
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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|
1763 <s xml:id="echoid-s154" 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/
casties
parents:
diff
changeset
|
1764 <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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|
1765 <s xml:id="echoid-s154" 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
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|
1766 S. W. R. was presumably Sir Walter Ralegh. |
|
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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|
1767 </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
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|
1768 </p> |
|
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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|
1769 </emph>] |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1770 <lb/><lb/></s></p></div> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff
changeset
|
1771 <p> |
|
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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|
1772 <s xml:id="echoid-s156" 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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parents:
diff
changeset
|
1773 S.W.R. <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
changeset
|
1774 L. Guy. <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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|
1775 L. Cobsam. <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
|
1776 M. G. Brooke. <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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|
1777 S. Ar. Gorge. <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
|
1778 S. Ar. Sanaher. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1779 S. Griffin Marcum |
|
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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|
1780 </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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|
1781 </p> |
|
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
|
1782 <pb file="add_6787_f112" o="112" n="222"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1783 <p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1784 <s xml:id="echoid-s157" 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
|
1785 At Rouhampton: : at Chalkhill <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1786 house <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1787 half of <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1788 party |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
1789 </s> |
|
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1790 </p> |
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1791 <pb file="add_6787_f112v" o="112v" n="223"/> |
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1792 <pb file="add_6787_f113" o="113" n="224"/> |
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1793 <p> |
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1794 <s xml:id="echoid-s158" xml:space="preserve"> |
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|
1795 Martins booke <lb/> |
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1796 Whitewell of the [???] <lb/> |
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1797 [???] of the vine <lb/> |
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1798 Tobacco |
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1799 </s> |
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1800 </p> |
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1801 <pb file="add_6787_f113v" o="113v" n="225"/> |
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1802 <pb file="add_6787_f114" o="114" n="226"/> |
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1803 <head xml:id="echoid-head37" xml:space="preserve" xml:lang="lat"> |
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1804 Archimedes. de cylindro <lb/> |
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1805 pa. 26. |
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1806 <lb/>[<emph style="it">tr: |
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|
1807 Archimedes, <emph style="it">De cylindro</emph>, page 26. |
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1808 </emph>]<lb/> |
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1809 </head> |
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1810 <pb file="add_6787_f114v" o="114v" n="227"/> |
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1811 <pb file="add_6787_f115" o="115" n="228"/> |
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1812 <head xml:id="echoid-head38" xml:space="preserve" xml:lang="lat"> |
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1813 Archimedes. de cylindro <lb/> |
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1814 pa. 19. |
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1815 <lb/>[<emph style="it">tr: |
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|
1816 Archimedes, <emph style="it">De cylindro</emph>, page 19. |
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1817 </emph>]<lb/> |
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1818 </head> |
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1819 <p xml:lang="lat"> |
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1820 <s xml:id="echoid-s159" xml:space="preserve"> |
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1821 pa. 19 |
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1822 <lb/>[<emph style="it">tr: |
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1823 page 19 |
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1824 </emph>]<lb/> |
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|
1825 </s> |
|
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|
1826 </p> |
|
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|
1827 <p xml:lang="lat"> |
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1828 <s xml:id="echoid-s160" xml:space="preserve"> |
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|
1829 coni igitur æquales |
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1830 <lb/>[<emph style="it">tr: |
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|
1831 therefore the cones are equal |
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|
1832 </emph>]<lb/> |
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|
1833 </s> |
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|
1834 </p> |
|
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|
1835 <p xml:lang="lat"> |
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|
1836 <s xml:id="echoid-s161" xml:space="preserve"> |
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|
1837 pa. 26. pro. 4 |
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1838 <lb/>[<emph style="it">tr: |
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|
1839 page 26, Proposition 4 |
|
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|
1840 </emph>]<lb/> |
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|
1841 </s> |
|
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|
1842 </p> |
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|
1843 <pb file="add_6787_f115v" o="115v" n="229"/> |
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|
1844 <pb file="add_6787_f116" o="116" n="230"/> |
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|
1845 <head xml:id="echoid-head39" xml:space="preserve" xml:lang="lat"> |
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|
1846 Archimedes. De cylindro. pag. 14. |
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|
1847 <lb/>[<emph style="it">tr: |
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|
1848 Archimedes, <emph style="it">De cylindro</emph>, page 14. |
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|
1849 </emph>]<lb/> |
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|
1850 </head> |
|
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|
1851 <p xml:lang="lat"> |
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|
1852 <s xml:id="echoid-s162" xml:space="preserve"> |
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|
1853 pa. 22 |
|
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|
1854 <lb/>[<emph style="it">tr: |
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|
1855 page 22 |
|
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|
1856 </emph>]<lb/> |
|
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|
1857 </s> |
|
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|
1858 </p> |
|
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|
1859 <p xml:lang="lat"> |
|
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|
1860 <s xml:id="echoid-s163" xml:space="preserve"> |
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|
1861 pa. 23 |
|
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|
1862 <lb/>[<emph style="it">tr: |
|
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|
1863 page 23 |
|
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|
1864 </emph>]<lb/> |
|
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|
1865 </s> |
|
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|
1866 </p> |
|
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|
1867 <p xml:lang="lat"> |
|
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|
1868 <s xml:id="echoid-s164" xml:space="preserve"> |
|
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|
1869 pa. 16 |
|
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|
1870 <lb/>[<emph style="it">tr: |
|
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|
1871 page 16 |
|
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|
1872 </emph>]<lb/> |
|
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|
1873 </s> |
|
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|
1874 </p> |
|
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|
1875 <p xml:lang="lat"> |
|
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|
1876 <s xml:id="echoid-s165" xml:space="preserve"> |
|
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|
1877 superficies <lb/> |
|
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|
1878 coni |
|
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|
1879 <lb/>[<emph style="it">tr: |
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1880 surface of a cone |
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1881 </emph>]<lb/> |
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1882 </s> |
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1883 <lb/> |
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1884 <s xml:id="echoid-s166" xml:space="preserve"> |
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1885 periferia circuli (a), et basis <lb/> |
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1886 coni eadem est |
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1887 <lb/>[<emph style="it">tr: |
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1888 the periphery of the circle (a) and the base of the cone are the same |
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1889 </emph>]<lb/> |
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1890 </s> |
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1891 </p> |
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1892 <p xml:lang="lat"> |
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1893 <s xml:id="echoid-s167" xml:space="preserve"> |
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1894 pa. 18 |
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1895 <lb/>[<emph style="it">tr: |
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1896 page 18 |
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1897 </emph>]<lb/> |
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1898 </s> |
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1899 </p> |
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1900 <pb file="add_6787_f116v" o="116v" n="231"/> |
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1901 <pb file="add_6787_f117" o="117" n="232"/> |
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1902 <pb file="add_6787_f117v" o="117v" n="233"/> |
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1903 <pb file="add_6787_f118" o="118" n="234"/> |
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1904 <pb file="add_6787_f118v" o="118v" n="235"/> |
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1905 <pb file="add_6787_f119" o="119" n="236"/> |
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1906 <pb file="add_6787_f119v" o="119v" n="237"/> |
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1907 <div xml:id="echoid-div39" type="page_commentary" level="2" n="39"> |
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1908 <p> |
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1909 <s xml:id="echoid-s168" xml:space="preserve">[<emph style="it">Note: |
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1910 <p> |
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1911 <s xml:id="echoid-s168" xml:space="preserve"> |
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1912 The verses referred to on this page are the following. |
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1913 </s> |
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1914 <lb/> |
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1915 <quote> |
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1916 Numbers 35.30: I any one kill a person, the murderer shall be put to death on the evidence of witnesses; |
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1917 but no person shall be put to death on the tetimony of one witness. |
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1918 </quote> |
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1919 <lb/> |
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1920 <quote> |
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1921 Deuteronomy 17.6: On the evidence of two witnesses or of three witnesses he that is to die shall be put to death; |
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1922 a person shall not be put to death on the evidence of one witness. |
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1923 </quote> |
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1924 <lb/> |
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1925 <quote> |
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1926 Deuteronomy 19.5: as when a man goes into the forest with his neighbour to cut wood, |
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1927 and his hand swings the axe to cut down the tree, and the head slips from the handle |
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1928 and strikes his neighbour so that he dies–he may flee to one of these cities and save his life; |
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1929 </quote> |
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1930 <lb/> |
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1931 <quote> |
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1932 Matthew 18.15–17: If your brother sins against you, go and tell him his fault, between you and him alone. |
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1933 If he listens to you, you have gained yoru brother. But if he does not listen, |
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1934 take one or two others along with you, that every word may be confirmed by the evidence of two or three witnesses. |
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1935 If he refuses to listen even to the church, let him be to you as a Gentile and a tax collector. |
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1936 </quote> |
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1937 <lb/> |
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1938 <quote> |
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1939 John 8.16–18: Yet even if I do judge, my judgement is true, for it is not I alone that judge, |
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1940 but I and he who sent me. In your law it is written that the testimony of two men is true; |
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1941 </quote> |
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1942 <lb/> |
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1943 <quote> |
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1944 2 Corinthians 13.1: This is the third time I am comoing to you. |
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1945 Any charge must be sustained by the evidence of two or three witnesses. |
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1946 </quote> |
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1947 <lb/> |
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1948 <quote> |
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1949 Matthew 7: 12: So whatever you wish that men would do to you, do so to them; for this is the law and the prophets. |
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1950 </quote> |
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1951 <lb/> |
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1952 <quote> |
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1953 Luke 6.13: And when it was day, he called his disciples, and chose from them twelve, whom he named apostles. |
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1954 </quote> |
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1955 <lb/> |
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1956 <quote> |
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1957 Tobit 4.15: And what you hate, do not do to anyone. Do not drink wine to excess or let drunkenness go with you on your way. |
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1958 </quote> |
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1959 <lb/> |
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1960 <quote> |
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1961 Matthew 7.2: For with the judgement you pronounce you will be judged, |
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1962 and the measure you give will be the the measure you get. |
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1963 </quote> |
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1964 <lb/> |
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1965 <quote> |
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1966 Proverbs 28.27: He who gives to the poor will not want, but he who hides his eyes will get many a curse. |
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1967 </quote> |
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1968 <lb/> |
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1969 <quote> |
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1970 Ecclesiasticus 27.26: Whoever digs a pit will fall into it, and whoever sets a snare will be caught in it. |
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1971 </quote> |
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1972 <lb/> |
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1973 <quote> |
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1974 Proverbs 19.5: A false witness will not go unpunished, and he who utters lies will not escape. |
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1975 </quote> |
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1976 </p> |
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1977 </emph>] |
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1978 <lb/><lb/></s></p></div> |
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1979 <p> |
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1980 <s xml:id="echoid-s170" xml:space="preserve"> |
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1981 Numbers. 35, 30. <lb/> |
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1982 Deut. 17, 6. <lb/> |
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1983 * Deut. 19, 5. <lb/> |
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1984 Math. 18, 15. 16, 17, <lb/> |
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1985 John. 8, 16. 17. 18. <lb/> |
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1986 2. Cor, 13, 1, <lb/> |
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1987 Math. 7, 12. <lb/> |
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1988 Luke. 6, 13. <lb/> |
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1989 Tobit. 4, 15, <lb/> |
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1990 Math. 7, 2. <lb/> |
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1991 prob. 28, 27. <lb/> |
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1992 Eccus. 27, 26. <lb/> |
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1993 prob. 19. 5 |
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1994 </s> |
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1995 </p> |
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1996 <pb file="add_6787_f120" o="120" n="238"/> |
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1997 <pb file="add_6787_f120v" o="120v" n="239"/> |
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1998 <pb file="add_6787_f121" o="121" n="240"/> |
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1999 <pb file="add_6787_f121v" o="121v" n="241"/> |
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2000 <pb file="add_6787_f122" o="122" n="242"/> |
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2001 <pb file="add_6787_f122v" o="122v" n="243"/> |
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2002 <pb file="add_6787_f123" o="123" n="244"/> |
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2003 <pb file="add_6787_f123v" o="123v" n="245"/> |
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2004 <pb file="add_6787_f124" o="124" n="246"/> |
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2005 <pb file="add_6787_f124v" o="124v" n="247"/> |
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2006 <pb file="add_6787_f125" o="125" n="248"/> |
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2007 <pb file="add_6787_f125v" o="125v" n="249"/> |
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2008 <pb file="add_6787_f126" o="126" n="250"/> |
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2009 <pb file="add_6787_f126v" o="126v" n="251"/> |
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2010 <pb file="add_6787_f127" o="127" n="252"/> |
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2011 <pb file="add_6787_f127v" o="127v" n="253"/> |
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2012 <pb file="add_6787_f128" o="128" n="254"/> |
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2013 <pb file="add_6787_f128v" o="128v" n="255"/> |
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2014 <pb file="add_6787_f129" o="129" n="256"/> |
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2015 <pb file="add_6787_f129v" o="129v" n="257"/> |
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2016 <pb file="add_6787_f130" o="130" n="258"/> |
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2017 <pb file="add_6787_f130v" o="130v" n="259"/> |
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2018 <pb file="add_6787_f131" o="131" n="260"/> |
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2019 <div xml:id="echoid-div40" type="page_commentary" level="2" n="40"> |
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2020 <p> |
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2021 <s xml:id="echoid-s171" xml:space="preserve">[<emph style="it">Note: |
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2022 <p> |
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2023 <s xml:id="echoid-s171" xml:space="preserve"> |
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2024 Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of |
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2025 Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). |
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2026 In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph> |
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2027 this triangle is to be found on page 421. |
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2028 </s> |
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2029 </p> |
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2030 </emph>] |
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2031 <lb/><lb/></s></p></div> |
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2032 <head xml:id="echoid-head40" xml:space="preserve" xml:lang="lat"> |
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2033 Vieta. resp. pag. 42, b. A. De Triangulis rectangulis sphaericis |
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2034 <lb/>[<emph style="it">tr: |
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2035 Viète, Responsorum liber VIII, page 42v. On right-angled spherical triangles. |
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2036 </emph>]<lb/> |
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2037 </head> |
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2038 <pb file="add_6787_f131v" o="131v" n="261"/> |
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2039 <pb file="add_6787_f132" o="132" n="262"/> |
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2040 <pb file="add_6787_f132v" o="132v" n="263"/> |
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2041 <pb file="add_6787_f133" o="133" n="264"/> |
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2042 <pb file="add_6787_f133v" o="133v" n="265"/> |
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2043 <pb file="add_6787_f134" o="134" n="266"/> |
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2044 <pb file="add_6787_f134v" o="134v" n="267"/> |
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2045 <pb file="add_6787_f135" o="135" n="268"/> |
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2046 <pb file="add_6787_f135v" o="135v" n="269"/> |
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2047 <pb file="add_6787_f136" o="136" n="270"/> |
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2048 <pb file="add_6787_f136v" o="136v" n="271"/> |
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2049 <pb file="add_6787_f137" o="137" n="272"/> |
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2050 <pb file="add_6787_f137v" o="137v" n="273"/> |
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2051 <pb file="add_6787_f138" o="138" n="274"/> |
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2052 <pb file="add_6787_f138v" o="138v" n="275"/> |
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2053 <pb file="add_6787_f139" o="139" n="276"/> |
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2054 <pb file="add_6787_f139v" o="139v" n="277"/> |
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2055 <pb file="add_6787_f140" o="140" n="278"/> |
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2056 <pb file="add_6787_f140v" o="140v" n="279"/> |
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2057 <pb file="add_6787_f141" o="141" n="280"/> |
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2058 <pb file="add_6787_f141v" o="141v" n="281"/> |
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2059 <pb file="add_6787_f142" o="142" n="282"/> |
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2060 <pb file="add_6787_f142v" o="142v" n="283"/> |
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2061 <pb file="add_6787_f143" o="143" n="284"/> |
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2062 <pb file="add_6787_f143v" o="143v" n="285"/> |
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2063 <pb file="add_6787_f144" o="144" n="286"/> |
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2064 <pb file="add_6787_f144v" o="144v" n="287"/> |
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2065 <pb file="add_6787_f145" o="145" n="288"/> |
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2066 <pb file="add_6787_f145v" o="145v" n="289"/> |
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2067 <pb file="add_6787_f146" o="146" n="290"/> |
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2068 <pb file="add_6787_f146v" o="146v" n="291"/> |
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2069 <pb file="add_6787_f147" o="147" n="292"/> |
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2070 <pb file="add_6787_f147v" o="147v" n="293"/> |
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2071 <pb file="add_6787_f148" o="148" n="294"/> |
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2072 <pb file="add_6787_f148v" o="148v" n="295"/> |
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2073 <pb file="add_6787_f149" o="149" n="296"/> |
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2074 <pb file="add_6787_f149v" o="149v" n="297"/> |
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2075 <pb file="add_6787_f150" o="150" n="298"/> |
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2076 <pb file="add_6787_f150v" o="150v" n="299"/> |
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2077 <pb file="add_6787_f151" o="151" n="300"/> |
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2078 <pb file="add_6787_f151v" o="151v" n="301"/> |
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2079 <pb file="add_6787_f152" o="152" n="302"/> |
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2081 <pb file="add_6787_f153" o="153" n="304"/> |
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2082 <pb file="add_6787_f153v" o="153v" n="305"/> |
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2083 <pb file="add_6787_f154" o="154" n="306"/> |
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2084 <pb file="add_6787_f154v" o="154v" n="307"/> |
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2085 <pb file="add_6787_f155" o="155" n="308"/> |
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2086 <pb file="add_6787_f155v" o="155v" n="309"/> |
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2087 <pb file="add_6787_f156" o="156" n="310"/> |
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2088 <pb file="add_6787_f156v" o="156v" n="311"/> |
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2089 <pb file="add_6787_f157" o="157" n="312"/> |
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2090 <pb file="add_6787_f157v" o="157v" n="313"/> |
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2091 <pb file="add_6787_f158" o="158" n="314"/> |
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2092 <pb file="add_6787_f158v" o="158v" n="315"/> |
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2093 <pb file="add_6787_f159" o="159" n="316"/> |
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2094 <pb file="add_6787_f159v" o="159v" n="317"/> |
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2095 <pb file="add_6787_f160" o="160" n="318"/> |
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2096 <div xml:id="echoid-div41" type="page_commentary" level="2" n="41"> |
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2097 <p> |
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2098 <s xml:id="echoid-s173" xml:space="preserve">[<emph style="it">Note: |
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2099 <p> |
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2100 <s xml:id="echoid-s173" xml:space="preserve"> |
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2101 At the end of Chapter XIX of Viète's |
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2102 <emph style="it">Variorum responsorum liber VIII</emph> (1593) |
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2103 there is a lengthy section entitled '<foreign xml:lang="gre">Eis procheiron scholia</foreign>. |
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2104 Section IV is on spherical geometry. The 18th and final proposition contains four statements, |
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2105 which Harriot here translated into symbolic notation. |
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2106 </s> |
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2107 <lb/> |
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2108 <quote xml:lang="lat"> |
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2109 18 Sit triangulum sphaericum ABD, & in peripheria BD cadat segmentum orthogonii AC. <lb/> |
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2110 Primo dico esse transsinuosa anguli BAC ad transsinuosa anguli DAC, |
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2111 sicut prosinum peripheria AB ad prosinum peripheri AD. <lb/> |
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2112 Secundo dico esse transsinuosam peripheriæ CB ad transsinuosam peripheriæ CD, |
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2113 sicut transsinuosam peripheriae AB ad transsinuosam peripheriæ AD. <lb/> |
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2114 Tertio dico esse sinum CD ad sinum CB, sicut prosinum anguli B ad prosinum anguli D. <lb/> |
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2115 Denique & quarto dico esse sinum anguli BAC ad sinum anguli DAC, |
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2116 sicut transsinuosam anguli D ad transsinuosam anguli B. |
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2117 </quote> |
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2118 <lb/> |
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2119 <quote> |
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2120 18. Let there be a spherical triangle ABD, and to the arc BD there falls an orthogonal line AC. <lb/> |
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2121 First I say that the secant of angle BAC to the secant of angle DAC is |
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2122 as the tangent of the arc AB to the tangent of the arc AD. <lb/> |
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2123 Second I say that the secant of the arc CB to the secant of the arc CD is |
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2124 as the secant of the arc AB to the secant of the arc AD. <lb/> |
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2125 Third I say that the sine of CD to the sine of DB is |
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2126 as the tangent of angle B to the tangent of angle D. <lb/> |
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2127 Fourth and last I say that the sine of angle BAC to the sine of angle DAC is |
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2128 as the secant of angle D to the secant of angle B. |
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2129 </quote> |
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2130 </p> |
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2131 </emph>] |
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2132 <lb/><lb/></s></p></div> |
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2133 <head xml:id="echoid-head41" xml:space="preserve" xml:lang="lat"> |
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2134 Demonstratio eorum quæ desiderant <lb/> |
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2135 in Vieta. lib. 8. respons. pag. 41.b. <lb/> |
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2136 sectione 18. |
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2137 <lb/>[<emph style="it">tr: |
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2138 Demonstration of what is missing in Viète, Responsorum liber VIII, page 41v, section 18. |
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2139 </emph>]<lb/> |
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2140 </head> |
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2141 <p xml:lang="lat"> |
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2142 <s xml:id="echoid-s175" xml:space="preserve"> |
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2143 Hinc apparet mendam <lb/> |
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2144 esse apud Vietam. nam <lb/> |
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2145 ille: |
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2146 <lb/>[<emph style="it">tr: |
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2147 Here is is clear that it is wrong in Viète, for this: |
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2148 </emph>]<lb/> |
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2149 </s> |
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2150 </p> |
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2151 <p xml:lang="lat"> |
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2152 <s xml:id="echoid-s176" xml:space="preserve"> |
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2153 Istæ quatuor conclusiones <lb/> |
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2154 sexeis variari possunt ut <lb/> |
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2155 ex analogijs rectangulorm est <lb/> |
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2156 manifestum. |
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2157 <lb/>[<emph style="it">tr: |
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2158 These four conclusions have six variations as is clear from the ratios for products. |
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2159 </emph>]<lb/> |
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2160 </s> |
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2161 </p> |
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2162 <pb file="add_6787_f160v" o="160v" n="319"/> |
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2163 <pb file="add_6787_f161" o="161" n="320"/> |
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2164 <div xml:id="echoid-div42" type="page_commentary" level="2" n="42"> |
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2165 <p> |
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2166 <s xml:id="echoid-s177" xml:space="preserve">[<emph style="it">Note: |
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2167 <p> |
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2168 <s xml:id="echoid-s177" xml:space="preserve"> |
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2169 Here Harriot examines a statement that appears as Proposition VI of the 'Dati sexti', Chapter XIX of |
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2170 Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). |
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2171 See also Add MS 6782, f. 439. |
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2172 </s> |
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2173 <lb/> |
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2174 <quote xml:lang="lat"> |
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2175 VI. <lb/> |
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2176 Data summa vel differentia duarum perpheriarum, quarum sinus datam habeant rationem, |
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2177 dantur singulares peripheriæ. |
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2178 </quote> |
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2179 <lb/> |
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2180 <quote> |
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2181 VI. Given the sum or difference of two arcs, whose sines are in a given ratio, each arc is given individually. |
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2182 </quote> |
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2183 <lb/> |
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2184 <s xml:id="echoid-s178" xml:space="preserve"> |
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2185 The reference to Pitiscus is to |
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2186 <emph style="it">Trigonometria: sive de solutione triangulorum tractarus brevis et perpsicuus</emph> (1595). |
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2187 </s> |
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2188 <lb/> |
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2189 <s xml:id="echoid-s179" xml:space="preserve"> |
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2190 The reference to Lansberg is to <emph style="it">Triangulorum geometriae libri quatuor</emph> (1591). |
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2191 </s> |
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2192 <lb/> |
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2193 <s xml:id="echoid-s180" xml:space="preserve"> |
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2194 The reference to Regiomontanus to <emph style="it">De triangulis omnimodis libri quinque</emph> ([1464], 1533, 1561), |
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2195 Book IV, Proposition 31. |
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2196 </s> |
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2197 </p> |
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2198 </emph>] |
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2199 <lb/><lb/></s></p></div> |
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2200 Data differentia.) <lb/> |
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2201 Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> differentia <emph style="super">duarum</emph> peripheriam. <lb/> |
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2202 ratio sinuum quæsitarum peripheriarum ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>. <lb/> |
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2203 <lb/>[...]<lb/> <lb/> |
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2204 sinus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi></mstyle></math> arcus <lb/> |
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2205 Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math>. nam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi><mo>+</mo><mi>e</mi><mi>c</mi><mo>=</mo><mi>d</mi><mi>e</mi></mstyle></math>. <lb/> |
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2206 Datur etiam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>o</mi></mstyle></math>. nam sinus complementi est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math> vel dimidij arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. <lb/> |
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2207 Cætera ut supra. et habetur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>k</mi></mstyle></math> sinus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi></mstyle></math> arcus. <lb/> |
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2208 Tum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi><mo>-</mo><mi>g</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>. arcus minor <lb/> |
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2209 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi><mo>+</mo><mi>g</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>. arcus maior quæsitis <lb/> |
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2210 Tum etiam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math> arcus maior quæsitis. |
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2211 <head xml:id="echoid-head42" xml:space="preserve" xml:lang="lat"> |
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2212 12.) Data summa vel differentia duarum periferiarum, <lb/> |
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2213 quarum sinus datam habeant rationem: dantur singulares <lb/> |
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2214 peripheriæ. |
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2215 <lb/>[<emph style="it">tr: |
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2216 Given the sum or difference of two arcs, for which the sines are in a given ratio, the individual arcs are given. |
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2217 </emph>]<lb/> |
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2218 </head> |
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2219 <p xml:lang="lat"> |
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2220 <s xml:id="echoid-s182" xml:space="preserve"> |
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2221 per Tangentes exhibit <lb/> |
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2222 Vieta in responsis pag. 37. <lb/> |
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2223 Pitiscus pag. 92. <lb/> |
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2224 Lansbergis. pag. 162 |
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2225 <lb/>[<emph style="it">tr: |
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2226 Shown by tangents <lb/> |
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2227 by Viète in Responsorum, page 37, <lb/> |
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2228 Pitiscus, page 92, <lb/> |
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2229 Lansberg, page 162. |
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2230 </emph>]<lb/> |
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2231 </s> |
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2232 </p> |
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2233 <p xml:lang="lat"> |
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2234 <s xml:id="echoid-s183" xml:space="preserve"> |
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2235 Quæ modo usui accomodatior est, quam per <lb/> |
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2236 sinus solos, quando tangentibus ut liceat. |
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2237 <lb/>[<emph style="it">tr: |
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2238 Which method of use is more convenient than by sines alone, when by tangents, as one pleases. |
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2239 </emph>]<lb/> |
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2240 </s> |
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2241 </p> |
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2242 <p xml:lang="lat"> |
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2243 <s xml:id="echoid-s184" xml:space="preserve"> |
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2244 Sed quando non licet Tangentibus uti, modus per solos sinus (etsi laboriosior) <lb/> |
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2245 adhibendus est. Exhibatur a Regiomontano lib. 4. prop. 31. de triangulis <lb/> |
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2246 Modus ille hic apponitur paucis explicatur. |
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2247 <lb/>[<emph style="it">tr: |
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2248 But when one does not want to use tangents, the method by sines alone (though more laborious) is shown. |
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2249 It is given by Regiomontanus, Book IV, Proposition 31 of De triangulis. |
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2250 That method set out here is explained a little. |
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2251 </emph>]<lb/> |
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2252 </s> |
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2253 </p> |
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2254 <p xml:lang="lat"> |
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2255 <s xml:id="echoid-s185" xml:space="preserve"> |
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2256 Data summa.) <lb/> |
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2257 Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> summa duarum peripheriam. <lb/> |
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2258 ratio sinuum quæsitarum peripheriarum ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>. <lb/> |
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2259 fiat: <lb/> |
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2260 Datur ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math>, nam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi><mo>-</mo><mi>e</mi><mi>c</mi><mo>=</mo><mi>d</mi><mi>e</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> est sinus dimidij arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. <lb/> |
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2261 Datur etiam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>o</mi></mstyle></math>. nam sinus complementi est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math> vel dimidij arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. <lb/> |
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2262 <lb/>[...]<lb/> <lb/> |
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2263 sinus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi></mstyle></math> arcus <lb/> |
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2264 Tum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi><mo>+</mo><mi>g</mi><mi>b</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math> arcus maior <lb/> |
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2265 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi><mo>-</mo><mi>g</mi><mi>b</mi><mo>=</mo><mi>b</mi><mi>c</mi></mstyle></math> arcus minor quæsita <lb/> |
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2266 Tum etiam: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>. arcus maior quæsitis. |
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2267 <lb/>[<emph style="it">tr: |
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2268 Given the sum.) <lb/> |
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2269 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> be the sum of the two arcs, |
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2270 and the ratio of the two sines of the sought arcs as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>. <lb/> |
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2271 construct: <lb/> |
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2272 Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math> is given, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi><mo>-</mo><mi>e</mi><mi>c</mi><mo>=</mo><mi>d</mi><mi>e</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> is the sine of half the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. <lb/> |
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2273 Also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>o</mi></mstyle></math> is gien, for the sine of the complement is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math>, or half the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. <lb/> |
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2274 <lb/>[...]<lb/> <lb/> |
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2275 the sine of arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi></mstyle></math> <lb/> |
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2276 Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi><mo>+</mo><mi>g</mi><mi>b</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>, the greater arc. <lb/> |
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2277 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi><mo>-</mo><mi>g</mi><mi>b</mi><mo>=</mo><mi>b</mi><mi>c</mi></mstyle></math> the lesser arc sought. <lb/> |
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2278 Then also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mo>-</mo><mi>b</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>, the greater arc sought. |
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2279 </emph>]<lb/> |
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2280 </s> |
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2281 </p> |
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2282 <p xml:lang="lat"> |
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2283 <s xml:id="echoid-s186" xml:space="preserve"> |
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2284 Data differentia.) <lb/> |
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2285 Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> differentia <emph style="super">duarum</emph> peripheriam. <lb/> |
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2286 ratio sinuum quæsitarum peripheriarum ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>. <lb/> |
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2287 <lb/>[...]<lb/> <lb/> |
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2288 sinus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi></mstyle></math> arcus <lb/> |
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2289 Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math>. nam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi><mo>+</mo><mi>e</mi><mi>c</mi><mo>=</mo><mi>d</mi><mi>e</mi></mstyle></math>. <lb/> |
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2290 Datur etiam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>o</mi></mstyle></math>. nam sinus complementi est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math> vel dimidij arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. <lb/> |
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2291 Cætera ut supra. et habetur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>k</mi></mstyle></math> sinus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi></mstyle></math> arcus. <lb/> |
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2292 Tum: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi><mo>-</mo><mi>g</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>. arcus minor <lb/> |
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2293 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi><mo>+</mo><mi>g</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>. arcus maior quæsitis <lb/> |
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2294 Tum etiam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math> arcus maior quæsitis. |
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2295 <lb/>[<emph style="it">tr: |
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2296 Given the difference.) <lb/> |
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2297 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> be the difference of the two sought arcs, |
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2298 and the ratio of the sines of the sought arcs <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>. <lb/> |
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2299 <lb/>[...]<lb/> <lb/> |
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2300 sine of the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi></mstyle></math> <lb/> |
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2301 Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math> is given, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi><mo>+</mo><mi>e</mi><mi>c</mi><mo>=</mo><mi>d</mi><mi>e</mi></mstyle></math>. <lb/> |
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2302 Also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>o</mi></mstyle></math> is given, for the sine of the complement is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>c</mi></mstyle></math>, or half the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math>. <lb/> |
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2303 The rest as above. And we have <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>k</mi></mstyle></math> the sine of arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi></mstyle></math>. <lb/> |
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2304 Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi><mo>-</mo><mi>g</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>, the lesser arc. <lb/> |
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2305 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi><mo>+</mo><mi>g</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>, the greater arc sought. <lb/> |
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2306 Then also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>=</mo><mi>a</mi><mi>b</mi></mstyle></math>, the greater arc sought. |
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2307 </emph>]<lb/> |
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2308 </s> |
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2309 </p> |
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2310 <pb file="add_6787_f161v" o="161v" n="321"/> |
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2311 <pb file="add_6787_f162" o="162" n="322"/> |
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2312 <head xml:id="echoid-head43" xml:space="preserve" xml:lang="lat"> |
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2313 13.) Data summa vel differentia <lb/> |
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2314 duarum peripheriarum, <lb/> |
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2315 quarum sinus datam habeant <lb/> |
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2316 rationem: dantur singulares <lb/> |
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2317 peripheriæ. |
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2318 <lb/>[<emph style="it">tr: |
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2319 Given the sum or difference of two arcs, for which the sines are in a given ratio, the individual arcs are given. |
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2320 </emph>]<lb/> |
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2321 </head> |
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2322 <pb file="add_6787_f162v" o="162v" n="323"/> |
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2323 <pb file="add_6787_f163" o="163" n="324"/> |
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2324 <pb file="add_6787_f163v" o="163v" n="325"/> |
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2325 <pb file="add_6787_f164" o="164" n="326"/> |
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2326 <pb file="add_6787_f164v" o="164v" n="327"/> |
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2327 <pb file="add_6787_f165" o="165" n="328"/> |
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2328 <pb file="add_6787_f165v" o="165v" n="329"/> |
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2329 <pb file="add_6787_f166" o="166" n="330"/> |
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2330 <pb file="add_6787_f166v" o="166v" n="331"/> |
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2331 <pb file="add_6787_f167" o="167" n="332"/> |
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2332 <div xml:id="echoid-div43" type="page_commentary" level="2" n="43"> |
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2333 <p> |
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2334 <s xml:id="echoid-s187" xml:space="preserve">[<emph style="it">Note: |
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2335 <p> |
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2336 <s xml:id="echoid-s187" xml:space="preserve"> |
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2337 This page refers to Propositions 48 and 49 of Book III of Apollonius, as edited by Commandino in |
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2338 <emph style="it">Conicorum libri quattuor</emph> (1566). |
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2339 </s> |
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2340 <lb/> |
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2341 <quote> |
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2342 III.48 |
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2343 With the same things being so, it must be shown that the straight lines drawn from the point of contact |
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2344 to the points produced by the application make equal angles with the tangent. |
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2345 </quote> |
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2346 <lb/> |
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2347 <quote> |
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2348 III.49 |
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2349 </quote> |
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2350 </p> |
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2351 </emph>] |
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2352 <lb/><lb/></s></p></div> |
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2353 <p xml:lang="lat"> |
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2354 <s xml:id="echoid-s189" xml:space="preserve"> |
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2355 Sit ellipsis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>f</mi><mi>k</mi></mstyle></math>: <lb/> |
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2356 cuius axis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>g</mi></mstyle></math> <lb/> |
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2357 centroides 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>w</mi></mstyle></math>. <lb/> |
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2358 diametroides, recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>w</mi></mstyle></math> <lb/> |
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2359 centrum, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <lb/> |
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2360 circulus circa axim, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>e</mi><mi>d</mi><mi>k</mi></mstyle></math>. <lb/> |
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2361 circulus circa diametroides, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>w</mi><mi>a</mi></mstyle></math>. <lb/> |
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2362 recta contingens ellipsin in <lb/> |
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2363 puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>d</mi></mstyle></math>. <lb/> |
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2364 perpendicularis a centroide <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math> <lb/> |
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2365 ad illam fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>e</mi></mstyle></math> <lb/> |
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2366 per 49.3 conicorum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>e</mi><mi>g</mi></mstyle></math> est angulus <lb/> |
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2367 rectus <lb/> |
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2368 ergo punctum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> in periferia. |
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2369 <lb/>[<emph style="it">tr: |
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2370 Let there be an ellipse <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>f</mi><mi>k</mi></mstyle></math> with axis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>g</mi></mstyle></math>, centroids at points <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>w</mi></mstyle></math>, diametroid the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>w</mi></mstyle></math>, |
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2371 centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. The circle about the axis is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>e</mi><mi>d</mi><mi>k</mi></mstyle></math>; the circle about the diametroid is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>w</mi><mi>a</mi></mstyle></math>; |
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2372 the line touching the ellipse at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi><mi>d</mi></mstyle></math>. Perpendicular to it from the centroid <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>, |
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2373 construct <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>e</mi></mstyle></math>. By Proposition III.49 of the Conics, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>e</mi><mi>g</mi></mstyle></math> is a right angle. Therefore, the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> |
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2374 is on the periphery. |
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2375 </emph>]<lb/> |
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2376 </s> |
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2377 </p> |
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2378 <p xml:lang="lat"> |
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2379 <s xml:id="echoid-s190" xml:space="preserve"> |
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2380 hinc sequitur <lb/> |
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2381 Si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>w</mi></mstyle></math> producatur ad periferium in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> <lb/> |
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2382 et ducatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>x</mi></mstyle></math> parallela ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math> <lb/> |
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2383 continget etiam ellipsin |
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2384 <lb/>[<emph style="it">tr: |
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2385 If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>w</mi></mstyle></math> is produced to the perpiphery at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>x</mi></mstyle></math> is taken paralle to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math>, it will also touch the ellipse. |
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2386 </emph>]<lb/> |
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2387 </s> |
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2388 </p> |
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2389 <p xml:lang="lat"> |
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2390 <s xml:id="echoid-s191" xml:space="preserve"> |
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2391 Si puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> in periferia <lb/> |
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2392 connectantur <lb/> |
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2393 linea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>x</mi></mstyle></math> transibit per alterum <lb/> |
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2394 centroides <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. |
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2395 <lb/>[<emph style="it">tr: |
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2396 If the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> in the periphery are joined, the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>x</mi></mstyle></math> will pass through the other centroid, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. |
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2397 </emph>]<lb/> |
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2398 </s> |
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2399 </p> |
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2400 <p xml:lang="lat"> |
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2401 <s xml:id="echoid-s192" xml:space="preserve"> |
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2402 Hinc. conclusio <lb/> |
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2403 Si circa <emph style="super">axim</emph> ellipseos describatur circulus <lb/> |
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2404 et in circulo inscribatur parallelogrammum <lb/> |
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2405 ita ut duo latera transeant per centroides: <lb/> |
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2406 reliqua duo contingent ellipsin. <lb/> |
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2407 et si duo latera contingent ellipsin; reliqua <lb/> |
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2408 duo transibunt per centroides. <lb/> |
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2409 Ita etam: <lb/> |
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2410 Si circa axim Hyperboles &c. |
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2411 <lb/>[<emph style="it">tr: |
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2412 Hence, the conclusion. <lb/> |
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2413 If around the axis of an ellipse there is described a circle, and in the circle there is inscribed a parallelogram |
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2414 so that two sides pass thorugh the centroids, the other two are tangents to the ellipse. |
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2415 And if two sides are tangents to the ellipse, the other two will pass throug the centroids. <lb/> |
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2416 Thus also: if around the axis of a hyperbola, etc. |
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2417 </emph>]<lb/> |
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2418 </s> |
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2419 </p> |
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2420 <p xml:lang="lat"> |
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2421 <s xml:id="echoid-s193" xml:space="preserve"> |
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2422 Alia conclusiones <lb/> |
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2423 iisdem positis. <lb/> |
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2424 per 48.3. conicorum. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>f</mi></mstyle></math> <lb/> |
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2425 faciunt æquale angulos ad <lb/> |
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2426 contingentem. <lb/> |
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2427 Si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>x</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>h</mi></mstyle></math> agantur <lb/> |
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2428 parallelæ ad contingentes: <lb/> |
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2429 puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> sunt in peri- <lb/> |
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2430 feria cuius diameter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>w</mi></mstyle></math>. |
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2431 <lb/>[<emph style="it">tr: |
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2432 Other conclusions form the same assumptions. <lb/> |
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2433 By Proposition III.48 of the Cinics, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>f</mi></mstyle></math> make equal angles to the tangent. <lb/> |
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2434 If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>x</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>h</mi></mstyle></math> are taken parallel to the tangents, the points <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>h</mi></mstyle></math> are on |
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2435 the circumference whose diamter is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>w</mi></mstyle></math>. |
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2436 </emph>]<lb/> |
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2437 </s> |
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2438 </p> |
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2439 <p xml:lang="lat"> |
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2440 <s xml:id="echoid-s194" xml:space="preserve"> |
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2441 Conveniat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>z</mi></mstyle></math> cum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>w</mi></mstyle></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>. <lb/> |
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2442 et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>h</mi></mstyle></math> cum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>a</mi></mstyle></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>v</mi></mstyle></math>. <lb/> |
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2443 Dico quod: <lb/> |
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2444 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>t</mi><mo>=</mo><mi>v</mi><mi>a</mi><mo>=</mo><mi>a</mi><mi>f</mi><mo>-</mo><mi>f</mi><mi>w</mi></mstyle></math> <lb/> |
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2445 nam: <lb/> |
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2446 anguli: |
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2447 <lb/>[<emph style="it">tr: |
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2448 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>z</mi></mstyle></math> meet with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>w</mi></mstyle></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>h</mi></mstyle></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>a</mi></mstyle></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>v</mi></mstyle></math>. I say that: <lb/> |
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2449 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi><mi>t</mi><mo>=</mo><mi>v</mi><mi>a</mi><mo>=</mo><mi>a</mi><mi>f</mi><mo>-</mo><mi>f</mi><mi>w</mi></mstyle></math> <lb/> |
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2450 for the angles: |
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2451 </emph>]<lb/> |
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2452 </s> |
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2453 </p> |
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2454 <p xml:lang="lat"> |
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2455 <s xml:id="echoid-s195" xml:space="preserve"> |
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2456 Dico etiam: |
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2457 <lb/>[<emph style="it">tr: |
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2458 I also say: |
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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="lat"> |
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2463 <s xml:id="echoid-s196" xml:space="preserve"> |
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2464 Dico etiam: |
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2465 <lb/>[<emph style="it">tr: |
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2466 I also say: |
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2467 </emph>]<lb/> |
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2468 </s> |
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2469 </p> |
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2470 <pb file="add_6787_f167v" o="167v" n="333"/> |
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2471 <pb file="add_6787_f168" o="168" n="334"/> |
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2472 <pb file="add_6787_f168v" o="168v" n="335"/> |
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2473 <pb file="add_6787_f169" o="169" n="336"/> |
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2474 <pb file="add_6787_f169v" o="169v" n="337"/> |
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2475 <pb file="add_6787_f170" o="170" n="338"/> |
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2476 <pb file="add_6787_f170v" o="170v" n="339"/> |
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2477 <pb file="add_6787_f171" o="171" n="340"/> |
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2478 <pb file="add_6787_f171v" o="171v" n="341"/> |
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2479 <pb file="add_6787_f172" o="172" n="342"/> |
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2480 <pb file="add_6787_f172v" o="172v" n="343"/> |
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2481 <pb file="add_6787_f173" o="173" n="344"/> |
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2482 <pb file="add_6787_f173v" o="173v" n="345"/> |
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2483 <pb file="add_6787_f174" o="174" n="346"/> |
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2484 <pb file="add_6787_f174v" o="174v" n="347"/> |
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2485 <pb file="add_6787_f175" o="175" n="348"/> |
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2486 <head xml:id="echoid-head44" xml:space="preserve" xml:lang="lat"> |
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2487 a.) 1.) De anomalijs Kepler. 284. 290. |
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2488 <lb/>[<emph style="it">tr: |
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2489 Kepler, De anomalijs, pages 284, 290. |
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2490 </emph>]<lb/> |
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2491 </head> |
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2492 <pb file="add_6787_f175v" o="175v" n="349"/> |
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2493 <pb file="add_6787_f176" o="176" n="350"/> |
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2494 <head xml:id="echoid-head45" xml:space="preserve" xml:lang="lat"> |
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2495 De anomalijs (a.1.) |
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2496 </head> |
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2497 <p xml:lang="lat"> |
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2498 <s xml:id="echoid-s197" xml:space="preserve"> |
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2499 Quod demonstrandum fuerit |
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2500 <lb/>[<emph style="it">tr: |
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|
2501 Which was to be proved |
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2502 </emph>]<lb/> |
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2503 </s> |
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2504 </p> |
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2505 <p xml:lang="lat"> |
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2506 <s xml:id="echoid-s198" xml:space="preserve"> |
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2507 4<emph style="super">o</emph>.) ut supra 5<emph style="super">o</emph> |
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2508 <lb/>[<emph style="it">tr: |
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2509 4 and the rest as above, and 5. |
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2510 </emph>]<lb/> |
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2511 </s> |
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2512 </p> |
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2513 <pb file="add_6787_f176v" o="176v" n="351"/> |
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2514 <pb file="add_6787_f177" o="177" n="352"/> |
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2515 <head xml:id="echoid-head46" xml:space="preserve" xml:lang="lat"> |
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2516 a.1 De anomalijs |
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2517 </head> |
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2518 <pb file="add_6787_f177v" o="177v" n="353"/> |
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2519 <pb file="add_6787_f178" o="178" n="354"/> |
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2520 <head xml:id="echoid-head47" xml:space="preserve" xml:lang="lat"> |
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2521 a.1) De anomalijs 284.290. |
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2522 </head> |
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2523 <pb file="add_6787_f178v" o="178v" n="355"/> |
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2524 <pb file="add_6787_f179" o="179" n="356"/> |
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2525 <head xml:id="echoid-head48" xml:space="preserve" xml:lang="lat"> |
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2526 a.1. Aliter et optime |
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2527 </head> |
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2528 <pb file="add_6787_f179v" o="179v" n="357"/> |
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2529 <pb file="add_6787_f180" o="180" n="358"/> |
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2530 <head xml:id="echoid-head49" xml:space="preserve" xml:lang="lat"> |
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2531 a.1. De anomalijs 299. |
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2532 </head> |
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2533 <pb file="add_6787_f180v" o="180v" n="359"/> |
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2534 <pb file="add_6787_f181" o="181" n="360"/> |
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2535 <head xml:id="echoid-head50" xml:space="preserve" xml:lang="lat"> |
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2536 a.1 De anomalijs Emendatur. <lb/> |
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2537 Keplerus. 299. |
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2538 <lb/>[<emph style="it">tr: |
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2539 De anomalijs. Amended. Kepler, page 299. |
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2540 </emph>]<lb/> |
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2541 </head> |
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2542 <pb file="add_6787_f181v" o="181v" n="361"/> |
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2543 <pb file="add_6787_f182" o="182" n="362"/> |
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2544 <head xml:id="echoid-head51" xml:space="preserve" xml:lang="lat"> |
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2545 a.2 De anomalijs |
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2546 </head> |
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2547 <pb file="add_6787_f182v" o="182v" n="363"/> |
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2548 <pb file="add_6787_f183" o="183" n="364"/> |
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2549 <head xml:id="echoid-head52" xml:space="preserve" xml:lang="lat"> |
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2550 a.3.) De anomalijs |
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2551 </head> |
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2552 <pb file="add_6787_f183v" o="183v" n="365"/> |
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2553 <pb file="add_6787_f184" o="184" n="366"/> |
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2554 <head xml:id="echoid-head53" xml:space="preserve" xml:lang="lat"> |
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2555 Kepler. pa. 290. |
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2556 </head> |
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2557 <pb file="add_6787_f184v" o="184v" n="367"/> |
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2558 <pb file="add_6787_f185" o="185" n="368"/> |
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2559 <pb file="add_6787_f185v" o="185v" n="369"/> |
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2560 <pb file="add_6787_f186" o="186" n="370"/> |
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2561 <pb file="add_6787_f186v" o="186v" n="371"/> |
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2562 <head xml:id="echoid-head54" xml:space="preserve" xml:lang="lat"> |
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2563 299. Kepr. |
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2564 <lb/>[<emph style="it">tr: |
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2565 Kepler, page 299. |
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2566 </emph>]<lb/> |
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2567 </head> |
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2568 <pb file="add_6787_f187" o="187" n="372"/> |
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2569 <head xml:id="echoid-head55" xml:space="preserve" xml:lang="lat"> |
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2570 299. <lb/> |
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2571 Charta. <lb/> |
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2572 (a.1) de <lb/> |
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2573 Anomalijs |
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2574 <lb/>[<emph style="it">tr: |
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2575 Page 299, Sheet (a.1), De anomalijs |
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2576 </emph>]<lb/> |
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2577 </head> |
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2578 <pb file="add_6787_f187v" o="187v" n="373"/> |
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2579 <pb file="add_6787_f188" o="188" n="374"/> |
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2580 <pb file="add_6787_f188v" o="188v" n="375"/> |
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2581 <pb file="add_6787_f189" o="189" n="376"/> |
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2582 <pb file="add_6787_f189v" o="189v" n="377"/> |
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2583 <pb file="add_6787_f190" o="190" n="378"/> |
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2584 <div xml:id="echoid-div44" type="page_commentary" level="2" n="44"> |
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2585 <p> |
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2586 <s xml:id="echoid-s199" xml:space="preserve">[<emph style="it">Note: |
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2587 <p> |
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2588 <s xml:id="echoid-s199" xml:space="preserve"> |
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2589 The text referred to here is Johan Philip Lansberg, |
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2590 <emph style="it">Triangulorum geometriae libri quatuor</emph> (1591). |
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2591 Lansberg 19s rules for spherical triangles are in Book 4, pages 167–207. |
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2592 </s> |
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2593 </p> |
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2594 </emph>] |
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2595 <lb/><lb/></s></p></div> |
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2596 <head xml:id="echoid-head56" xml:space="preserve" xml:lang="lat"> |
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|
2597 Anapherosis trianguli obliquanguli <lb/> |
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|
2598 Lansbergi |
|
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2599 <lb/>[<emph style="it">tr: |
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|
2600 Anapherosis for oblique-angled triangles, from Lansberg |
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2601 </emph>]<lb/> |
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2602 </head> |
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2603 <p xml:lang="lat"> |
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2604 conversio |
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2605 <lb/>[<emph style="it">tr: |
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|
2606 conversion |
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2607 </emph>]<lb/> |
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2608 </p> |
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2609 <pb file="add_6787_f190v" o="190v" n="379"/> |
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2610 <pb file="add_6787_f191" o="191" n="380"/> |
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2611 <div xml:id="echoid-div45" type="page_commentary" level="2" n="45"> |
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2612 <p> |
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2613 <s xml:id="echoid-s201" xml:space="preserve">[<emph style="it">Note: |
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2614 <p> |
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2615 <s xml:id="echoid-s201" xml:space="preserve"> |
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|
2616 Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of |
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|
2617 Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). |
|
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|
2618 In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph>, |
|
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|
2619 diagrams related to the one drawn here are to be found on pages 423 and 424. |
|
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|
2620 </s> |
|
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2621 </p> |
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|
2622 </emph>] |
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2623 <lb/><lb/></s></p></div> |
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2624 <head xml:id="echoid-head57" xml:space="preserve" xml:lang="lat"> |
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|
2625 Anapherosis Trianguli Vietani |
|
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|
2626 <lb/>[<emph style="it">tr: |
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|
2627 Anapherosis in a triangle of Viète |
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|
2628 </emph>]<lb/> |
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|
2629 </head> |
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|
2630 <pb file="add_6787_f191v" o="191v" n="381"/> |
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|
2631 <pb file="add_6787_f192" o="192" n="382"/> |
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|
2632 <pb file="add_6787_f192v" o="192v" n="383"/> |
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|
2633 <pb file="add_6787_f193" o="193" n="384"/> |
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|
2634 <div xml:id="echoid-div46" type="page_commentary" level="2" n="46"> |
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|
2635 <p> |
|
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|
2636 <s xml:id="echoid-s203" xml:space="preserve">[<emph style="it">Note: |
|
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|
2637 <p> |
|
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|
2638 <s xml:id="echoid-s203" xml:space="preserve"> |
|
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|
2639 Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of |
|
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|
2640 Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). |
|
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|
2641 In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph>, |
|
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|
2642 diagrams related to the one drawn here are to be found on pages 423 and 424. |
|
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|
2643 </s> |
|
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|
2644 </p> |
|
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|
2645 </emph>] |
|
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|
2646 <lb/><lb/></s></p></div> |
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|
2647 <pb file="add_6787_f193v" o="193v" n="385"/> |
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|
2648 <pb file="add_6787_f194" o="194" n="386"/> |
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|
2649 <div xml:id="echoid-div47" type="page_commentary" level="2" n="47"> |
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|
2650 <p> |
|
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|
2651 <s xml:id="echoid-s205" xml:space="preserve">[<emph style="it">Note: |
|
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|
2652 <p> |
|
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|
2653 <s xml:id="echoid-s205" xml:space="preserve"> |
|
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|
2654 Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of |
|
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|
2655 Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). |
|
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|
2656 In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph>, |
|
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|
2657 diagrams related to the one drawn here are to be found on pages 423 and 424. |
|
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|
2658 </s> |
|
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|
2659 </p> |
|
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|
2660 </emph>] |
|
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|
2661 <lb/><lb/></s></p></div> |
|
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|
2662 <head xml:id="echoid-head58" xml:space="preserve" xml:lang="lat"> |
|
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|
2663 Conversio triangul Vietani |
|
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|
2664 <lb/>[<emph style="it">tr: |
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|
2665 Conversion of Viète's triangle |
|
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|
2666 </emph>]<lb/> |
|
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|
2667 </head> |
|
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|
2668 <pb file="add_6787_f194v" o="194v" n="387"/> |
|
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|
2669 <pb file="add_6787_f195" o="195" n="388"/> |
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|
2670 <div xml:id="echoid-div48" type="page_commentary" level="2" n="48"> |
|
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|
2671 <p> |
|
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|
2672 <s xml:id="echoid-s207" xml:space="preserve">[<emph style="it">Note: |
|
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|
2673 <p> |
|
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|
2674 <s xml:id="echoid-s207" xml:space="preserve"> |
|
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|
2675 Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of |
|
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|
2676 Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). |
|
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|
2677 In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph>, |
|
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|
2678 the diagram drawn here is to be found on pages 423 and 424. |
|
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|
2679 </s> |
|
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|
2680 </p> |
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2681 </emph>] |
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2682 <lb/><lb/></s></p></div> |
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2683 <pb file="add_6787_f195v" o="195v" n="389"/> |
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2684 <pb file="add_6787_f196" o="196" n="390"/> |
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2685 <div xml:id="echoid-div49" type="page_commentary" level="2" n="49"> |
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2686 <p> |
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2687 <s xml:id="echoid-s209" xml:space="preserve">[<emph style="it">Note: |
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2688 <p> |
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2689 <s xml:id="echoid-s209" xml:space="preserve"> |
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2690 Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of |
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2691 Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). |
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2692 In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph>, |
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2693 diagrams related to the one drawn here are to be found on pages 421 to 426. |
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2694 </s> |
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2695 </p> |
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2696 </emph>] |
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2697 <lb/><lb/></s></p></div> |
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2698 <pb file="add_6787_f196v" o="196v" n="391"/> |
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2699 <pb file="add_6787_f197" o="197" n="392"/> |
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2700 <div xml:id="echoid-div50" type="page_commentary" level="2" n="50"> |
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2701 <p> |
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2702 <s xml:id="echoid-s211" xml:space="preserve">[<emph style="it">Note: |
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2703 <p> |
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2704 <s xml:id="echoid-s211" xml:space="preserve"> |
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2705 The text referred to here is Johan Philip Lansberg, |
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2706 <emph style="it">Triangulorum geometriae libri quatuor</emph> (1591). |
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2707 Lansberg 19s rule for finding a side of a spherical triangles, given its angles, |
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2708 is the final rule in Book 4, on page 201. |
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2709 </s> |
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2710 </p> |
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2711 </emph>] |
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2712 <lb/><lb/></s></p></div> |
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2713 <head xml:id="echoid-head59" xml:space="preserve" xml:lang="lat"> |
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2714 Datis tribus angulis quaesitur latus. Lansbergi trianguli<lb/> |
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2715 solutio. |
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2716 <lb/>[<emph style="it">tr: |
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2717 Given three angles, a side is sought. Solution from Lansberg, Triangulorum geometriae. |
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2718 </emph>]<lb/> |
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2719 </head> |
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2720 <pb file="add_6787_f197v" o="197v" n="393"/> |
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2721 <pb file="add_6787_f198" o="198" n="394"/> |
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2722 <div xml:id="echoid-div51" type="page_commentary" level="2" n="51"> |
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2723 <p> |
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2724 <s xml:id="echoid-s213" xml:space="preserve">[<emph style="it">Note: |
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2725 <p> |
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2726 <s xml:id="echoid-s213" xml:space="preserve"> |
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2727 Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of |
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2728 Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). |
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2729 The table relates to the various diagrams in Add MS 6787, f. 191 to f. 195. |
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2730 </s> |
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2731 </p> |
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2732 </emph>] |
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2733 <lb/><lb/></s></p></div> |
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2734 <head xml:id="echoid-head60" xml:space="preserve" xml:lang="lat"> |
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2735 De Anapherosi et conversione traingulorum |
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2736 <lb/>[<emph style="it">tr: |
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2737 On Anapherosis and the conversion of triangles |
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2738 </emph>]<lb/> |
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2739 </head> |
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2740 <pb file="add_6787_f198v" o="198v" n="395"/> |
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2741 <pb file="add_6787_f199" o="199" n="396"/> |
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2742 <div xml:id="echoid-div52" type="page_commentary" level="2" n="52"> |
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2743 <p> |
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2744 <s xml:id="echoid-s215" xml:space="preserve">[<emph style="it">Note: |
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2745 <p> |
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2746 <s xml:id="echoid-s215" xml:space="preserve"> |
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|
2747 Further work on the '<foreign xml:lang="gre">Eis procheiron scholia</foreign>, which follows Chapter XIX of |
|
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|
2748 Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). |
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|
2749 In the 1646 edition of Viete's <emph style="it">Opera mathematica</emph> |
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2750 the triangle referred to here is to be found on page 424. |
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2751 </s> |
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2752 </p> |
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2753 </emph>] |
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2754 <lb/><lb/></s></p></div> |
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2755 <head xml:id="echoid-head61" xml:space="preserve" xml:lang="lat"> |
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|
2756 Datis tribus lateri, quaeruntur anguli |
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2757 <lb/>[<emph style="it">tr: |
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|
2758 Given three sides, the angles the angles are sought. |
|
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|
2759 </emph>]<lb/> |
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|
2760 </head> |
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|
2761 <p xml:lang="lat"> |
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|
2762 <s xml:id="echoid-s217" xml:space="preserve"> |
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|
2763 Quæritur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> |
|
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|
2764 <lb/>[<emph style="it">tr: |
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|
2765 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> is sought |
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2766 </emph>]<lb/> |
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2767 </s> |
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2768 </p> |
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2769 <pb file="add_6787_f199v" o="199v" n="397"/> |
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2770 <pb file="add_6787_f200" o="200" n="398"/> |
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2771 <p> |
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2772 <s xml:id="echoid-s218" xml:space="preserve"> |
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2773 Wernerus <emph style="st">discipulus Regiomontani</emph><lb/> |
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2774 Duiditius <lb/> |
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2775 Mr Savill <lb/> |
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2776 Mr Warner |
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2777 </s> |
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2778 </p> |
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2779 <pb file="add_6787_f200v" o="200v" n="399"/> |
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2780 <pb file="add_6787_f201" o="201" n="400"/> |
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2781 <pb file="add_6787_f201v" o="201v" n="401"/> |
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2782 <pb file="add_6787_f202" o="202" n="402"/> |
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2783 <pb file="add_6787_f202v" o="202v" n="403"/> |
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2784 <pb file="add_6787_f203" o="203" n="404"/> |
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2785 <pb file="add_6787_f203v" o="203v" n="405"/> |
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2786 <pb file="add_6787_f204" o="204" n="406"/> |
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2787 <pb file="add_6787_f204v" o="204v" n="407"/> |
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2788 <pb file="add_6787_f205" o="205" n="408"/> |
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2789 <pb file="add_6787_f205v" o="205v" n="409"/> |
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2790 <pb file="add_6787_f206" o="206" n="410"/> |
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2791 <pb file="add_6787_f206v" o="206v" n="411"/> |
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2792 <pb file="add_6787_f207" o="207" n="412"/> |
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2793 <pb file="add_6787_f207v" o="207v" n="413"/> |
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2794 <pb file="add_6787_f208" o="208" n="414"/> |
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2795 <div xml:id="echoid-div53" type="page_commentary" level="2" n="53"> |
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2796 <p> |
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2797 <s xml:id="echoid-s219" xml:space="preserve">[<emph style="it">Note: |
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2798 <p> |
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2799 <s xml:id="echoid-s219" xml:space="preserve"> |
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2800 The reference on this page is to Viète's |
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2801 <emph style="it">Variorum responsorum liber VIII</emph>, Chapter 13, |
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2802 entitled 'Angulus cornicularis'. |
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2803 There Viète gave 6 arguments about the angle between the tangent and circumference of a circle. |
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2804 Harriot has converted each of them into ratios, written in the lower half of the page in symbolic notation. |
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2805 </s> |
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2806 </p> |
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2807 </emph>] |
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2808 <lb/><lb/></s></p></div> |
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2809 <head xml:id="echoid-head62" xml:space="preserve"> |
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2810 Cornus. Vieta habet 6 sequentes analogias |
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2811 <lb/>[<emph style="it">tr: |
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2812 Horn angle. Viète has the 6 following ratios. |
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2813 </emph>]<lb/> |
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2814 </head> |
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2815 <pb file="add_6787_f208v" o="208v" n="415"/> |
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2816 <pb file="add_6787_f209" o="209" n="416"/> |
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2817 <div xml:id="echoid-div54" type="page_commentary" level="2" n="54"> |
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2818 <p> |
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2819 <s xml:id="echoid-s221" xml:space="preserve">[<emph style="it">Note: |
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2820 <p> |
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2821 <s xml:id="echoid-s221" xml:space="preserve"> |
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|
2822 This page appears to be a continuation of Add MS 6787, f. 208, |
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2823 concerned with Viète's arguments about the horn angle in |
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|
2824 <emph style="it">Variorum responsorum liber VIII</emph>, Chapter 13. |
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2825 </s> |
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2826 </p> |
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2827 </emph>] |
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2828 <lb/><lb/></s></p></div> |
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2829 <head xml:id="echoid-head63" xml:space="preserve"> |
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2830 De Triangulis Sphæricis Rectangulis <lb/> |
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2831 Cornus |
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2832 <lb/>[<emph style="it">tr: |
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|
2833 On right-angles spherical triangles. Horn angle. |
|
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2834 </emph>]<lb/> |
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2835 </head> |
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2836 <pb file="add_6787_f209v" o="209v" n="417"/> |
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|
2837 <pb file="add_6787_f210" o="210" n="418"/> |
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|
2838 <div xml:id="echoid-div55" type="page_commentary" level="2" n="55"> |
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|
2839 <p> |
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2840 <s xml:id="echoid-s223" xml:space="preserve">[<emph style="it">Note: |
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2841 <p> |
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2842 <s xml:id="echoid-s223" xml:space="preserve"> |
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|
2843 This page appears to be a continuation of Add MS 6787, f. 208 and f. 209, |
|
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|
2844 concerned with Viète's arguments about the horn angle in |
|
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|
2845 <emph style="it">Variorum responsorum liber VIII</emph>, Chapter 13. |
|
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2846 </s> |
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2847 </p> |
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2848 </emph>] |
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|
2849 <lb/><lb/></s></p></div> |
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|
2850 <head xml:id="echoid-head64" xml:space="preserve"> |
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|
2851 Siphon |
|
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2852 </head> |
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2853 <p xml:lang="lat"> |
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2854 <s xml:id="echoid-s225" xml:space="preserve"> |
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2855 Ista 6 analogiæ sunt <lb/> |
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2856 apud vieta. |
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2857 <lb/>[<emph style="it">tr: |
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2858 These 6 ratios are in Viète. |
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2859 </emph>]<lb/> |
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2860 </s> |
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2861 </p> |
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2862 <pb file="add_6787_f210v" o="210v" n="419"/> |
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2863 <pb file="add_6787_f211" o="211" n="420"/> |
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2864 <pb file="add_6787_f211v" o="211v" n="421"/> |
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2865 <pb file="add_6787_f212" o="212" n="422"/> |
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2866 <pb file="add_6787_f212v" o="212v" n="423"/> |
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2867 <pb file="add_6787_f213" o="213" n="424"/> |
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2868 <pb file="add_6787_f213v" o="213v" n="425"/> |
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2869 <pb file="add_6787_f214" o="214" n="426"/> |
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2870 <pb file="add_6787_f214v" o="214v" n="427"/> |
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2871 <pb file="add_6787_f215" o="215" n="428"/> |
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2872 <pb file="add_6787_f215v" o="215v" n="429"/> |
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2873 <pb file="add_6787_f216" o="216" n="430"/> |
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2874 <pb file="add_6787_f216v" o="216v" n="431"/> |
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2875 <pb file="add_6787_f217" o="217" n="432"/> |
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2876 <pb file="add_6787_f217v" o="217v" n="433"/> |
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2877 <pb file="add_6787_f218" o="218" n="434"/> |
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2878 <pb file="add_6787_f218v" o="218v" n="435"/> |
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2879 <pb file="add_6787_f219" o="219" n="436"/> |
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2880 <pb file="add_6787_f219v" o="219v" n="437"/> |
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2881 <pb file="add_6787_f220" o="220" n="438"/> |
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2882 <pb file="add_6787_f220v" o="220v" n="439"/> |
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2883 <pb file="add_6787_f221" o="221" n="440"/> |
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2884 <div xml:id="echoid-div56" type="page_commentary" level="2" n="56"> |
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2885 <p> |
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2886 <s xml:id="echoid-s226" xml:space="preserve">[<emph style="it">Note: |
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2887 <p> |
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2888 <s xml:id="echoid-s226" xml:space="preserve"> |
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2889 At the end of Chapter XIX of <emph style="it">Variorum responsorum liber VIII</emph> (1593), |
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2890 under the heading 'ALIUD', Viète listed sixteen propositions connecting sines, tangents, and secants. |
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2891 In the first edition of the <emph style="it">Responsorum</emph>, the pages are numbered only on the recto side. |
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2892 However, the pagination goes badly wrong, so that in Chapter XIX we have the sequence: |
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2893 37, 38, 39, 38. This sometimes makes it difficult to follow Harriot's references correctly. |
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2894 Here it seems that he has seen the number '38' on the right-hand (recto) page, |
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2895 and thus inferred that the left-hand (verso) page must be 37v, whereas it is in face 39v. |
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2896 In the 1646 edition of Viète's <emph style="it">Opera mathematica</emph> |
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2897 the sixteen propositions are to be found on pages 412–413. |
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2898 </s> |
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2899 <lb/> |
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2900 <s xml:id="echoid-s227" xml:space="preserve"> |
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2901 On this and the following pages, Harriot worked through the sixteen propositions systematically. |
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2902 On this page he lists the first six. |
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2903 Note that for Harriot, as for Viète, trigonometrical relationships arose from astronomy. |
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2904 Thus the concepts of sine, tangent, and secant related not to angles defined by a pair of lines meeting at a point, |
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2905 but to arcs of a circle with a given radius, and therefore only by implication to the angles subtended by them. |
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2906 </s> |
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2907 <lb/> |
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2908 <s xml:id="echoid-s228" xml:space="preserve"> |
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2909 At the top of the page Harriot listed the relevant quantities: |
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2910 sine, tangent, secant, radius (or whole sine), and the symbols he regularly used for them. |
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2911 The equivalent names used by Viète were sinus, prosinus, transinuosa, totus. |
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2912 Harriot had no words for cosine, cotangent, or coseceant. |
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2913 Where we would use 'cosine', for example, he spoke of the sine of the complement. |
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2914 Thus he wrote <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> BC for sine(arc BC) |
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2915 but <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>υ</mi></mstyle></math> <emph style="st">BC</emph> for the sine of the complement of BC, that is, cosine(BC). |
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2916 </s> |
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2917 </p> |
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2918 </emph>] |
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2919 <lb/><lb/></s></p></div> |
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2920 <head xml:id="echoid-head65" xml:space="preserve" xml:lang="lat"> |
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2921 vieta in lib. 8. respons. <lb/> |
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2922 pag. 37. <lb/> |
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2923 proportionalia |
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2924 <lb/>[<emph style="it">tr: |
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2925 Viète, in Responsorum liber VIII, page 37, proportionals |
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2926 </emph>]<lb/> |
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2927 </head> |
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2928 <p xml:lang="lat"> |
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2929 <s xml:id="echoid-s230" xml:space="preserve"> |
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2930 sinus <lb/> |
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2931 tangens <lb/> |
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2932 secans <lb/> |
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2933 radius |
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2934 <lb/>[<emph style="it">tr: |
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2935 sine <lb/> |
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2936 tangent <lb/> |
|
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2937 secans <lb/> |
|
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2938 radius |
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2939 </emph>]<lb/> |
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2940 </s> |
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2941 </p> |
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2942 <p xml:lang="lat"> |
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2943 <s xml:id="echoid-s231" xml:space="preserve"> |
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|
2944 1. Sinus peripheriæ, Radius: Radius, Secans complementi. <lb/> |
|
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2945 2. Sinus comp. peripheriæ. Radius. Radius. Secans peripheriæ. <lb/> |
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2946 3. Tangens peripheriæ. Radius. Radius. Tangens complementi. |
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2947 <lb/>[<emph style="it">tr: |
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2948 1. Sine of the arc : Radius = Radius : Secant of the complement. <lb/> |
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2949 2. Sine of the compplement of the arc : Radius = Radius : Secant of the arc. <lb/> |
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2950 3. Tangent of the arc : Radius = Radius : Tangent of the complement. |
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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-s232" xml:space="preserve"> |
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2956 Ergo: cum proponuntur duæ peripheriæ <lb/> |
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2957 4. Sinus peripheriæ primæ. Sinus secundæ. Secans. compl. secundæ. Secans compl. primæ. <lb/> |
|
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2958 5. Sinus compl. primæ. Sinus com. secundæ. Secans 2<emph style="super">æ</emph>. Secans primæ. <lb/> |
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2959 6. Tangens primæ. Tangens 2<emph style="super">æ</emph>. |
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2960 Tangens comp. 2<emph style="super">æ</emph>. Tangens compl. primæ |
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2961 <lb/>[<emph style="it">tr: |
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2962 Therefore, when there are given two arcs: <lb/> |
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2963 4. Sine of the first arc : Sine of the second = |
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2964 Secant of the complement of the second : Secant of the complement of the first. <lb/> |
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2965 5. Sine of the complement of the first : SIne of the complement of the second = |
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|
2966 Secant of the second : Secant of the first. <lb/> |
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2967 6. Tangent of the first : Tangent of the second = |
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2968 Tangent of the complemnt of the second : Tangent of the complement of the first. |
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2969 </emph>]<lb/> |
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|
2970 </s> |
|
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2971 </p> |
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2972 <p xml:lang="lat"> |
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2973 <s xml:id="echoid-s233" xml:space="preserve"> |
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|
2974 Menda in Vieta. <lb/> |
|
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|
2975 correctio. |
|
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2976 <lb/>[<emph style="it">tr: |
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2977 Wrong in Viète. <lb/> |
|
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|
2978 Correction. </emph>]<lb/> |
|
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|
2979 </s> |
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|
2980 </p> |
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2981 <p xml:lang="lat"> |
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2982 <s xml:id="echoid-s234" xml:space="preserve"> |
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|
2983 sinus <lb/> |
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|
2984 tangens <lb/> |
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2985 secans <lb/> |
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2986 radius |
|
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2987 </s> |
|
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2988 </p> |
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2989 <pb file="add_6787_f221v" o="221v" n="441"/> |
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|
2990 <pb file="add_6787_f222" o="222" n="442"/> |
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2991 <div xml:id="echoid-div57" type="page_commentary" level="2" n="57"> |
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2992 <p> |
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2993 <s xml:id="echoid-s235" xml:space="preserve">[<emph style="it">Note: |
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2994 <p> |
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2995 <s xml:id="echoid-s235" xml:space="preserve"> |
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|
2996 On this page, a continuation from Add MS 6787, f. 221, Harriot lists and proves Propositions 7 to 11 |
|
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|
2997 from the 'ALIUD' in Chapter XIX of Viète's <emph style="it">Variorum resposorum liber VIII</emph>. |
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2998 </s> |
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2999 </p> |
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|
3000 </emph>] |
|
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|
3001 <lb/><lb/></s></p></div> |
|
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|
3002 <head xml:id="echoid-head66" xml:space="preserve" xml:lang="lat"> |
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|
3003 Vieta lib. 8. resp. <lb/> |
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|
3004 pag. 37 |
|
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3005 <lb/>[<emph style="it">tr: |
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|
3006 Viète, in Responsorum liber VIII, page 37. |
|
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|
3007 </emph>]<lb/> |
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3008 </head> |
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3009 <p xml:lang="lat"> |
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3010 <s xml:id="echoid-s237" xml:space="preserve"> |
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|
3011 Lemma, satis evidens ex elementis: <lb/> |
|
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|
3012 ad demonstrandam 7<emph style="super">am</emph> prop. <lb/> |
|
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|
3013 1. |
|
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3014 <lb/>[<emph style="it">tr: |
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|
3015 Lemma, sufficiently evident from the fundamentals; |
|
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|
3016 used for the demonstration of the 7th proposition. <lb/> |
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|
3017 1. |
|
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|
3018 </emph>]<lb/> |
|
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|
3019 </s> |
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|
3020 </p> |
|
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|
3021 <p xml:lang="lat"> |
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|
3022 <s xml:id="echoid-s238" xml:space="preserve"> |
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|
3023 Ergo pro 7<emph style="super">a</emph>. |
|
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|
3024 <lb/>[<emph style="it">tr: |
|
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|
3025 Therefore for the 7th proposition |
|
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|
3026 </emph>]<lb/> |
|
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|
3027 </s> |
|
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|
3028 </p> |
|
22d6a63640c6
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|
3029 <p xml:lang="lat"> |
|
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|
3030 <s xml:id="echoid-s239" xml:space="preserve"> |
|
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|
3031 2. Alia dispositio terminorum lemmatis <lb/> |
|
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|
3032 ad demonstrandam 7<emph style="super">am</emph> prop. |
|
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|
3033 <lb/>[<emph style="it">tr: |
|
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|
3034 2. Another arrangement of the terms in the lemma. |
|
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|
3035 </emph>]<lb/> |
|
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|
3036 </s> |
|
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|
3037 </p> |
|
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|
3038 <p xml:lang="lat"> |
|
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|
3039 <s xml:id="echoid-s240" xml:space="preserve"> |
|
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|
3040 Ergo pro 8<emph style="super">a</emph>. |
|
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|
3041 <lb/>[<emph style="it">tr: |
|
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|
3042 Therefore for the 8th proposition. |
|
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|
3043 </emph>]<lb/> |
|
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|
3044 </s> |
|
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|
3045 </p> |
|
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|
3046 <p xml:lang="lat"> |
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|
3047 <s xml:id="echoid-s241" xml:space="preserve"> |
|
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|
3048 per 5tam <lb/> |
|
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|
3049 Et alterne <lb/> |
|
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|
3050 Ergo <lb/> |
|
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|
3051 Et alterne |
|
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3052 <lb/>[<emph style="it">tr: |
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|
3053 By the 5th <lb/> |
|
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|
3054 And alternately <lb/> |
|
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|
3055 Therefore <lb/> |
|
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|
3056 And alternately |
|
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|
3057 </emph>]<lb/> |
|
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|
3058 </s> |
|
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|
3059 </p> |
|
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|
3060 <p xml:lang="lat"> |
|
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|
3061 <s xml:id="echoid-s242" xml:space="preserve"> |
|
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|
3062 Ergo pro 9<emph style="super">a</emph>: per 7<emph style="super">am</emph>. |
|
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|
3063 <lb/>[<emph style="it">tr: |
|
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|
3064 Therefore for the 9th; by the 7th. |
|
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|
3065 </emph>]<lb/> |
|
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|
3066 </s> |
|
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|
3067 </p> |
|
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|
3068 <p xml:lang="lat"> |
|
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|
3069 <s xml:id="echoid-s243" xml:space="preserve"> |
|
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|
3070 3. Lemmatis variato |
|
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|
3071 <lb/>[<emph style="it">tr: |
|
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|
3072 3. A variation of the lemmma. |
|
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|
3073 </emph>]<lb/> |
|
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|
3074 </s> |
|
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|
3075 </p> |
|
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|
3076 <p xml:lang="lat"> |
|
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|
3077 <s xml:id="echoid-s244" xml:space="preserve"> |
|
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|
3078 Ergo pro 10<emph style="super">a</emph>: per 7<emph style="super">am</emph>. <lb/> |
|
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|
3079 per 6<emph style="super">a</emph> alterne. |
|
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|
3080 <lb/>[<emph style="it">tr: |
|
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|
3081 Therefore for the 10th; by the 7th. |
|
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|
3082 </emph>]<lb/> |
|
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|
3083 </s> |
|
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|
3084 </p> |
|
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|
3085 <p xml:lang="lat"> |
|
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|
3086 <s xml:id="echoid-s245" xml:space="preserve"> |
|
22d6a63640c6
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|
3087 4. Lemmatis variato |
|
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|
3088 <lb/>[<emph style="it">tr: |
|
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|
3089 4. A variation of the lemma. |
|
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|
3090 </emph>]<lb/> |
|
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|
3091 </s> |
|
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|
3092 </p> |
|
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|
3093 <p xml:lang="lat"> |
|
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|
3094 <s xml:id="echoid-s246" xml:space="preserve"> |
|
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|
3095 Ergo pro 11<emph style="super">a</emph>: per 7<emph style="super">am</emph>. <lb/> |
|
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|
3096 per 5<emph style="super">a</emph> alterne. |
|
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|
3097 <lb/>[<emph style="it">tr: |
|
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|
3098 Therefore for the 11th; by the 7th. |
|
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|
3099 </emph>]<lb/> |
|
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|
3100 </s> |
|
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|
3101 </p> |
|
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|
3102 <pb file="add_6787_f222v" o="222v" n="443"/> |
|
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|
3103 <pb file="add_6787_f223" o="223" n="444"/> |
|
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|
3104 <div xml:id="echoid-div58" type="page_commentary" level="2" n="58"> |
|
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|
3105 <p> |
|
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|
3106 <s xml:id="echoid-s247" xml:space="preserve">[<emph style="it">Note: |
|
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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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|
3107 <p> |
|
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|
3108 <s xml:id="echoid-s247" xml:space="preserve"> |
|
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|
3109 On this page, a continuation from Add MS 6787, f. 221, f. 222, and f. 225, |
|
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|
3110 Harriot lists and proves Propositions 13 to 16 from the 'ALIUD' in Chapter XIX of Viète's |
|
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|
3111 <emph style="it">Variorum resposorum liber VIII</emph>. |
|
22d6a63640c6
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|
3112 </s> |
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3113 <lb/> |
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3114 <s xml:id="echoid-s248" xml:space="preserve"> |
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3115 The reference to Regiomontanus to <emph style="it">De triangulis omnimodis libri quinque</emph> ([1464], 1533, 1561), |
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3116 Book V, Proposition 1. |
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3117 </s> |
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3118 <lb/> |
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3119 <s xml:id="echoid-s249" xml:space="preserve"> |
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3120 The reference to Fink is to <emph style="it">Geometriae rotundi libri XIIII</emph> (1583), page 364. |
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3121 </s> |
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3122 </p> |
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3123 </emph>] |
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3124 <lb/><lb/></s></p></div> |
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3125 <head xml:id="echoid-head67" xml:space="preserve" xml:lang="lat"> |
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3126 Vieta. lib. 8. responsorum. <lb/> |
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3127 pag. 37b. |
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3128 <lb/>[<emph style="it">tr: |
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3129 Viète, in Responsorum liber VIII, page 37. |
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3130 </emph>]<lb/> |
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3131 </head> |
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3132 <p xml:lang="lat"> |
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3133 <s xml:id="echoid-s251" xml:space="preserve"> |
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3134 Et cum proponuntur tres peripheriæ. <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>A</mi><mi>C</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math>. |
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3135 <lb/>[<emph style="it">tr: |
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3136 And when there are given three arcs <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>A</mi><mi>C</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math>. |
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3137 </emph>]<lb/> |
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3138 </s> |
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3139 </p> |
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3140 <p xml:lang="lat"> |
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3141 <s xml:id="echoid-s252" xml:space="preserve"> |
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3142 Superiores Analogiæ demonstrantur <lb/> |
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3143 per istas quæ sequntur. |
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3144 <lb/>[<emph style="it">tr: |
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3145 The above ratios may be demonstrated by these which follow. |
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3146 </emph>]<lb/> |
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3147 </s> |
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3148 </p> |
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3149 <p xml:lang="lat"> |
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3150 <s xml:id="echoid-s253" xml:space="preserve"> |
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3151 [???] 13 habetur in Reg. lib. 5. 1. <lb/> |
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3152 et Finkio pag. 364. <lb/> |
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3153 Notavi in alia charta <lb/> |
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3154 [???] |
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3155 <lb/>[<emph style="it">tr: |
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3156 [???] 13 is to be had in Regiomontanus. Book V, Proposition 1, and Fink, page 364. <lb/> |
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3157 I have noted in another sheet [???]. |
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3158 </emph>]<lb/> |
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3159 </s> |
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3160 </p> |
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3161 <pb file="add_6787_f223v" o="223v" n="445"/> |
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3162 <pb file="add_6787_f224" o="224" n="446"/> |
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3163 <div xml:id="echoid-div59" type="page_commentary" level="2" n="59"> |
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3164 <p> |
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3165 <s xml:id="echoid-s254" xml:space="preserve">[<emph style="it">Note: |
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3166 <p> |
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3167 <s xml:id="echoid-s254" xml:space="preserve"> |
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3168 At the end of Chapter XIX of Viète's |
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3169 <emph style="it">Variorum responsorum liber VIII</emph> (1593) |
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3170 there is a lengthy section entitled '<foreign xml:lang="gre">Eis procheiron scholia</foreign>. |
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3171 Section IV is on spherical geometry. The 18th and final proposition contains four statements, |
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3172 which Harriot here translated into symbolic notation. |
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3173 </s> |
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3174 <lb/> |
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3175 <quote xml:lang="lat"> |
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3176 18 Sit triangulum sphaericum ABD, & in peripheria BD cadat segmentum orthogonii AC. <lb/> |
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3177 Primo dico esse transsinuosa anguli BAC ad transsinuosa anguli DAC, |
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3178 sicut prosinum peripheria AB ad prosinum peripheri AD. <lb/> |
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3179 Secundo dico esse transsinuosam peripheriæ CB ad transsinuosam peripheriæ CD, |
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3180 sicut transsinuosam peripheriae AB ad transsinuosam peripheriæ AD. <lb/> |
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3181 Tertio dico esse sinum CD ad sinum CB, sicut prosinum anguli B ad prosinum anguli D. <lb/> |
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3182 Denique & quarto dico esse sinum anguli BAC ad sinum anguli DAC, |
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3183 sicut transsinuosam anguli D ad transsinuosam anguli B. |
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3184 </quote> |
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3185 <lb/> |
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3186 <quote> |
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3187 18. Let there be a spherical triangle ABD, and to the arc BD there falls an orthogonal line AC. <lb/> |
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3188 First I say that the secant of angle BAC to the secant of angle DAC is |
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3189 as the tangent of the arc AB to the tangent of the arc AD. <lb/> |
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3190 Second I say that the secant of the arc CB to the secant of the arc CD is |
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3191 as the secant of the arc AB to the secant of the arc AD. <lb/> |
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3192 Third I say that the sine of CD to the sine of DB is |
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3193 as the tangent of angle B to the tangent of angle D. <lb/> |
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3194 Fourth and last I say that the sine of angle BAC to the sine of angle DAC is |
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3195 as the secant of angle D to the secant of angle B. |
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3196 </quote> |
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3197 </p> |
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3198 </emph>] |
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3199 <lb/><lb/></s></p></div> |
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3200 <head xml:id="echoid-head68" xml:space="preserve" xml:lang="lat"> |
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3201 Vieta lib. 8. resp. <lb/> |
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3202 pag. 41. b. <lb/> |
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3203 prop. 18 |
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3204 <lb/>[<emph style="it">tr: |
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3205 Viète, Responsorum liber VIII, page 41v, Proposition 18. |
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3206 </emph>]<lb/> |
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3207 </head> |
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3208 <p xml:lang="lat"> |
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3209 <s xml:id="echoid-s256" xml:space="preserve"> |
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3210 Hinc apparet mendam <lb/> |
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3211 esse apud Vietam. |
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3212 <lb/>[<emph style="it">tr: |
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3213 Here is is clear that it is wrong in Viète. |
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3214 </emph>]<lb/> |
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3215 </s> |
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3216 </p> |
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3217 <pb file="add_6787_f224v" o="224v" n="447"/> |
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3218 <pb file="add_6787_f225" o="225" n="448"/> |
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3219 <div xml:id="echoid-div60" type="page_commentary" level="2" n="60"> |
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3220 <p> |
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3221 <s xml:id="echoid-s257" xml:space="preserve">[<emph style="it">Note: |
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3222 <p> |
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3223 <s xml:id="echoid-s257" xml:space="preserve"> |
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3224 On this page, a continuation from Add MS 6787, f. 221 and f. 222, |
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3225 Harriot gives a diagram for the proof of Proposition 12 |
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3226 from the 'ALIUD' in Chapter XIX of Viète's <emph style="it">Variorum resposorum liber VIII</emph>. |
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3227 </s> |
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3228 </p> |
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3229 </emph>] |
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3230 <lb/><lb/></s></p></div> |
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3231 <head xml:id="echoid-head69" xml:space="preserve" xml:lang="lat"> |
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3232 Vieta. lib. 8. responsorum. <lb/> |
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3233 pag. 37b. <lb/> |
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3234 Diagramma ad demonstrationem prop. 12. |
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3235 <lb/>[<emph style="it">tr: |
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3236 Viète, in Responsorum liber VIII, page 37. <lb/> |
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3237 Diagram for the demonstration of Proposition 12. |
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3238 </emph>]<lb/> |
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3239 </head> |
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3240 <p xml:lang="lat"> |
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3241 <s xml:id="echoid-s259" xml:space="preserve"> |
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3242 Manifesta ex diagrammate |
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3243 <lb/>[<emph style="it">tr: |
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3244 Obvious from the diagram. |
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3245 </emph>]<lb/> |
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3246 </s> |
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3247 </p> |
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3248 <p xml:lang="lat"> |
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3249 <s xml:id="echoid-s260" xml:space="preserve"> |
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|
3250 Anaologiæ eædem in notis universalibus. |
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3251 <lb/>[<emph style="it">tr: |
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|
3252 The same ratios in general notation. |
|
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3253 </emph>]<lb/> |
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3254 </s> |
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3255 </p> |
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3256 <p xml:lang="lat"> |
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3257 <s xml:id="echoid-s261" xml:space="preserve"> |
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|
3258 Ergo pro 12<emph style="super">a</emph>. |
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3259 <lb/>[<emph style="it">tr: |
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|
3260 Therefore for the 12th. |
|
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|
3261 </emph>]<lb/> |
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|
3262 </s> |
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3263 </p> |
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3264 <p xml:lang="lat"> |
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3265 <s xml:id="echoid-s262" xml:space="preserve"> |
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|
3266 Hinc notatur Menda in Vieta <lb/> |
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|
3267 et emendatur. |
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3268 <lb/>[<emph style="it">tr: |
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|
3269 Here it is noted that it is wrong in Viète, and amended. |
|
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3270 </emph>]<lb/> |
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3271 </s> |
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3272 </p> |
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|
3273 <pb file="add_6787_f225v" o="225v" n="449"/> |
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|
3274 <pb file="add_6787_f226" o="226" n="450"/> |
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3275 <div xml:id="echoid-div61" type="page_commentary" level="2" n="61"> |
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3276 <p> |
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3277 <s xml:id="echoid-s263" xml:space="preserve">[<emph style="it">Note: |
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3278 <p> |
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3279 <s xml:id="echoid-s263" xml:space="preserve"> |
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|
3280 The first three propositions from the 'ALIUD' in Chapter XIX of Viète's |
|
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3281 <emph style="it">Variorum resposorum liber VIII</emph>, |
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|
3282 written in symbolic notation. |
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3283 </s> |
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3284 </p> |
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|
3285 </emph>] |
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3286 <lb/><lb/></s></p></div> |
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3287 <pb file="add_6787_f226v" o="226v" n="451"/> |
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3288 <pb file="add_6787_f227" o="227" n="452"/> |
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3289 <div xml:id="echoid-div62" type="page_commentary" level="2" n="62"> |
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3290 <p> |
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3291 <s xml:id="echoid-s265" xml:space="preserve">[<emph style="it">Note: |
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3292 <p> |
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3293 <s xml:id="echoid-s265" xml:space="preserve"> |
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3294 This is the first of four pages devoted to Proposition 14 from Chapter XIX of Viète's |
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3295 <emph style="it">Variorum responsorum liber VIII</emph>, |
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3296 a lengthy chapter on plane and spherical triangles. |
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3297 </s> |
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3298 <lb/> |
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3299 <quote xml:lang="lat"> |
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3300 XIV. <lb/> |
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3301 Prothechidion. <lb/> |
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3302 Data duorum maximorum in sphæra circulorum inclinatione, quorum unus secatur a tertio per alterius polos, |
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3303 arguitur quanta fit maxima differentia suarum a nodo longitudinum. <lb/> |
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3304 Et contra. Ex maxima differentia longitudinum a nodo, arguitur quanta fit circulum inclinatio. |
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3305 </quote> |
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3306 <lb/> |
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3307 <quote> |
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3308 Given the inclination of two great circles on a sphere, |
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3309 one of which is cut by a third through the pole of the other, |
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3310 there is to be found the greatest difference in their longitudes from the node. |
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3311 Conversely, from the greatest difference of longitudes from the node, |
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3312 there may be found the inclination of the circles. |
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3313 </quote> |
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3314 <s xml:id="echoid-s266" xml:space="preserve"> |
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3315 In the crossed out sentence halfway down the page there are references to Fink and Clavius. <lb/> |
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3316 The reference to Fink is to his <emph style="it">Geometriae rotundi libri XIIII</emph> (1583). <lb/> |
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3317 The reference to Clavius is to his <emph style="it">Triangula rectilinea, atque sphaerica</emph> (1586). |
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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-head70" xml:space="preserve"> |
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3323 Vieta. lib. 8. resp. |
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3324 pag. 35. prop. 14. <foreign xml:lang="gre">proch?on</foreign> |
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3325 <lb/>[<emph style="it">tr: |
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3326 Viète, Responsorum liber VIII, page 35, Proposition 14, |
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3327 </emph>]<lb/> |
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3328 </head> |
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3329 <p xml:lang="lat"> |
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3330 <s xml:id="echoid-s268" xml:space="preserve"> |
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3331 Ista propositio est utilis in calcu-<lb/> |
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3332 lationibus astronimicis. per eam cog-<lb/> |
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3333 noscitur maxima differentia inter <lb/> |
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3334 numerationes per Eclipticam et proprios <lb/> |
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3335 circulos planetorum, et ubi est &c. <lb/> |
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3336 Etiam: <lb/> |
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3337 æquationibus <lb/> |
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3338 dierum. |
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3339 <lb/>[<emph style="it">tr: |
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3340 This proposition is useful in astronomical calculations. |
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3341 By it may be known the maximum difference between observations by the ecliptic and the nearest orbits of planets, |
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3342 and where it is. <lb/> |
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3343 Also, the equations of the days. |
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3344 </emph>]<lb/> |
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3345 </s> |
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3346 </p> |
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3347 <p xml:lang="lat"> |
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3348 <s xml:id="echoid-s269" xml:space="preserve"> |
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3349 Sit triangulum rectangulum <lb/> |
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3350 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math>, angulus rectus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>. Quæritur <lb/> |
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3351 Maxima differentia inter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> et <lb/> |
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3352 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math>. Nam arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>B</mi><mi>C</mi></mstyle></math> in diversis <lb/> |
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3353 positionibus 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>E</mi></mstyle></math>, facit diversis <lb/> |
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3354 differentias longitudinum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> et <lb/> |
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3355 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math>. |
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3356 <lb/>[<emph style="it">tr: |
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3357 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math> be a right-angled triangle with right angle at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>. |
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3358 There is sought the maximum differene between <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>A</mi><mi>C</mi></mstyle></math>. |
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3359 For the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>B</mi><mi>C</mi></mstyle></math> in various positions 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>E</mi></mstyle></math> |
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3360 makes various differences of longitude <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>A</mi><mi>C</mi></mstyle></math>. |
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3361 </emph>]<lb/> |
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3362 </s> |
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3363 </p> |
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3364 <p xml:lang="lat"> |
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3365 <s xml:id="echoid-s270" xml:space="preserve"> |
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3366 polo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>. <emph style="super">et</emph> per punctum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> describatur <lb/> |
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3367 parallelus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi><mi>B</mi><mi>D</mi></mstyle></math>. Ergo: <lb/> |
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3368 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> est differentia inter <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>A</mi><mi>C</mi></mstyle></math> <lb/> |
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3369 sit diameter paralleli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>O</mi></mstyle></math> et <lb/> |
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3370 sit perpendcularis illi, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>M</mi></mstyle></math>. |
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3371 <lb/>[<emph style="it">tr: |
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3372 Taking the pole <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, there is drawn through <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> parallel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi><mi>B</mi><mi>D</mi></mstyle></math>. <lb/> |
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3373 Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> is the difference between <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>A</mi><mi>C</mi></mstyle></math>. |
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3374 Let the diameter parallel to it be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>O</mi></mstyle></math> and the perpendicuar to it <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>M</mi></mstyle></math>. |
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3375 </emph>]<lb/> |
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3376 </s> |
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3377 </p> |
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3378 <p xml:lang="lat"> |
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3379 <s xml:id="echoid-s271" xml:space="preserve"> |
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3380 <emph style="st"> |
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3381 Dico quando <emph style="super">*</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>M</mi></mstyle></math> linea est æqualis sinui <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math>, hoc est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>. Tum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi><mi>O</mi></mstyle></math> erit æqualis <lb/> |
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3382 semidiametro sphæræ, scilicet <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>D</mi><mi>C</mi></mstyle></math> erit differentia quæsita maxima. <lb/> |
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3383 Pro demonstratione nota diagramma in Finkio pag. 393. et Clavium 445. |
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3384 </emph> |
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3385 <lb/>[<emph style="it">tr: |
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3386 I say that when the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>M</mi></mstyle></math> is equal to the sine <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math>, that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>, |
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3387 then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi><mi>O</mi></mstyle></math> will be equal to the semidiameter of a spehre, namely <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>F</mi></mstyle></math>, |
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3388 and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> will be the sought maximum difference. |
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3389 </emph>]<lb/> |
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3390 </s> |
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3391 </p> |
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3392 <p xml:lang="lat"> |
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3393 <s xml:id="echoid-s272" xml:space="preserve"> |
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3394 Hic notabo solummodo proportiones in Vieta. <lb/> |
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3395 Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>I</mi></mstyle></math>, sinus arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>F</mi></mstyle></math>, hoc est anguli <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>F</mi><mi>I</mi></mstyle></math> erit sinus versus [???]. <lb/> |
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3396 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>D</mi></mstyle></math> est arcus similis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>F</mi></mstyle></math>. Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>K</mi><mi>L</mi></mstyle></math> æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>I</mi></mstyle></math>. et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi><mi>O</mi><mo>=</mo><mi>D</mi><mi>M</mi></mstyle></math>. cætera |
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3397 <emph style="st">ex</emph> <emph style="super">in</emph> diagramata. |
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3398 <lb/>[<emph style="it">tr: |
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3399 Here I have noted only the proportions in Viète. |
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3400 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>I</mi></mstyle></math>, be the sine of arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>F</mi></mstyle></math>, that is of angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>. The versed sine will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>I</mi></mstyle></math>. |
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3401 The arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>D</mi></mstyle></math> is similar to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>F</mi></mstyle></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>K</mi><mi>L</mi></mstyle></math> be equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>I</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi><mi>O</mi><mo>=</mo><mi>D</mi><mi>M</mi></mstyle></math>. |
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3402 The rest from the diagram. |
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3403 </emph>]<lb/> |
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3404 </s> |
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3405 </p> |
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3406 <p xml:lang="lat"> |
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3407 <s xml:id="echoid-s273" xml:space="preserve"> |
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3408 * <lb/> |
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3409 Dico quando <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>O</mi></mstyle></math> habet <lb/> |
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3410 <emph style="st">ratio</emph> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math> sinum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math>: <lb/> |
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3411 eandem rationem quam <lb/> |
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3412 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi><mi>M</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi><mi>D</mi></mstyle></math> vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>L</mi><mi>I</mi></mstyle></math> ad <lb/> |
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3413 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>F</mi></mstyle></math>. Tum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> erit <lb/> |
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3414 differentia maxima. <lb/> |
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3415 Vel melius ita: <lb/> |
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3416 Quando <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>O</mi></mstyle></math> sit <lb/> |
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3417 parallela <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>. hoc <lb/> |
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3418 est quando <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>H</mi><mi>O</mi></mstyle></math> est <lb/> |
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3419 rectus angulus. <lb/> |
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3420 Demonstratio <lb/> |
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3421 Habetur in alia <lb/> |
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3422 charta annexa. |
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3423 <lb/>[<emph style="it">tr: |
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3424 I say that when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>O</mi></mstyle></math> has the same ratio to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>, the sine of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi><mi>M</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi><mi>D</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>L</mi><mi>I</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>F</mi></mstyle></math>, |
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3425 then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> will be the maximum difference. <lb/> |
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3426 Or better thus: when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>O</mi></mstyle></math> is a right angle. <lb/> |
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3427 The demonstration is to be found in another sheet, adjoined. |
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3428 </emph>]<lb/> |
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3429 </s> |
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3430 </p> |
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3431 <pb file="add_6787_f227v" o="227v" n="453"/> |
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3432 <pb file="add_6787_f228" o="228" n="454"/> |
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3433 <div xml:id="echoid-div63" type="page_commentary" level="2" n="63"> |
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3434 <p> |
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3435 <s xml:id="echoid-s274" xml:space="preserve">[<emph style="it">Note: |
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3436 <p> |
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3437 <s xml:id="echoid-s274" xml:space="preserve"> |
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3438 Further work on Proposition 14 from Chapter XIX of Viète's |
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3439 <emph style="it">Variorum responsorum liber VIII</emph>, |
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3440 continued from Add MS 6787, f. 227. |
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3441 </s> |
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3442 </p> |
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3443 </emph>] |
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3444 <lb/><lb/></s></p></div> |
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3445 <head xml:id="echoid-head71" xml:space="preserve"> |
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3446 Vieta. lib. 8. resp. |
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3447 pag. 35. prop. 14. <foreign xml:lang="gre">proch?on</foreign> |
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3448 <lb/>[<emph style="it">tr: |
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3449 Viète, Responsorum liber VIII, page 35, Proposition 14, |
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3450 </emph>]<lb/> |
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3451 </head> |
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3452 <pb file="add_6787_f228v" o="228v" n="455"/> |
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3453 <pb file="add_6787_f229" o="229" n="456"/> |
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3454 <div xml:id="echoid-div64" type="page_commentary" level="2" n="64"> |
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3455 <p> |
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3456 <s xml:id="echoid-s276" xml:space="preserve">[<emph style="it">Note: |
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3457 <p> |
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3458 <s xml:id="echoid-s276" xml:space="preserve"> |
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3459 Further work on Proposition 14 from Chapter XIX of Viète's |
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3460 <emph style="it">Variorum responsorum liber VIII</emph>, |
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3461 continued from Add MS 6787, f. 227 and f. 228. |
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3462 </s> |
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3463 </p> |
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3464 </emph>] |
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3465 <lb/><lb/></s></p></div> |
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3466 <head xml:id="echoid-head72" xml:space="preserve"> |
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3467 Vieta. lib. 8. resp. |
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3468 pag. 35. prop. 14. <foreign xml:lang="gre">proch?on</foreign> |
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3469 <lb/>[<emph style="it">tr: |
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3470 Viète, Responsorum liber VIII, page 35, Proposition 14, |
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3471 </emph>]<lb/> |
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3472 </head> |
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3473 <p xml:lang="lat"> |
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3474 <s xml:id="echoid-s278" xml:space="preserve"> |
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3475 Habita maxima differentia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> et eius sinu <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>: sinus arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>, <lb/> |
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3476 hoc est linea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>Q</mi></mstyle></math> ita invenietur. <lb/> |
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3477 <omision/> <lb/> |
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3478 Ergo nota <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>M</mi></mstyle></math> <lb/> |
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3479 <lb/>[...]<lb/> <lb/> |
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3480 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>Q</mi></mstyle></math> quæsita |
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3481 <lb/>[<emph style="it">tr: |
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3482 Having the maximum difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> and its sine <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>, |
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3483 then the sine of the arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>, that is, the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>Q</mi></mstyle></math> is found thus. <lb/> |
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3484 <lb/>[...]<lb/> <lb/> |
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3485 Therefore note <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>M</mi></mstyle></math> <lb/> |
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3486 <lb/>[...]<lb/> <lb/> |
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3487 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>Q</mi></mstyle></math>, the line sought |
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3488 </emph>]<lb/> |
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3489 </s> |
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3490 </p> |
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3491 <p xml:lang="lat"> |
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3492 <s xml:id="echoid-s279" xml:space="preserve"> |
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3493 vel ita Brevius <lb/> |
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3494 In triangulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math>, cum etiam datur angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, <lb/> |
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3495 per doctrinam triangulorum dabitur <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>A</mi><mi>C</mi></mstyle></math>. |
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3496 <lb/>[<emph style="it">tr: |
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3497 In the triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math>, since the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> is also given, |
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3498 by the teaching on triangles there will be given <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>A</mi><mi>C</mi></mstyle></math>. |
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3499 </emph>]<lb/> |
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3500 </s> |
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3501 </p> |
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3502 <pb file="add_6787_f229v" o="229v" n="457"/> |
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3503 <pb file="add_6787_f230" o="230" n="458"/> |
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3504 <div xml:id="echoid-div65" type="page_commentary" level="2" n="65"> |
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3505 <p> |
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3506 <s xml:id="echoid-s280" xml:space="preserve">[<emph style="it">Note: |
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3507 <p> |
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3508 <s xml:id="echoid-s280" xml:space="preserve"> |
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3509 Further work on Proposition 14 from Chapter XIX of Viète's |
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3510 <emph style="it">Variorum responsorum liber VIII</emph>, |
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3511 continued from Add MS 6787, f. 227, f. 228, and f. 229. |
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3512 </s> |
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3513 </p> |
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3514 </emph>] |
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3515 <lb/><lb/></s></p></div> |
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3516 <head xml:id="echoid-head73" xml:space="preserve"> |
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3517 Vieta. lib. 8. resp. |
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3518 pag. 35. prop. 14. <foreign xml:lang="gre">proch?on</foreign> |
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3519 <lb/>[<emph style="it">tr: |
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3520 Viète, Responsorum liber VIII, page 35, Proposition 14, |
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3521 </emph>]<lb/> |
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3522 </head> |
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3523 <p xml:lang="lat"> |
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3524 <s xml:id="echoid-s282" xml:space="preserve"> |
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3525 Synopsis Demonstrationis illius proportionis. |
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3526 <lb/>[<emph style="it">tr: |
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3527 Sysnopsis of the demonstration of this proportion. |
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3528 </emph>]<lb/> |
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3529 </s> |
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3530 </p> |
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3531 <p xml:lang="lat"> |
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3532 <s xml:id="echoid-s283" xml:space="preserve"> |
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3533 sunt in eadem ratione |
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3534 <lb/>[<emph style="it">tr: |
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3535 are in the same ratio |
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3536 </emph>]<lb/> |
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3537 </s> |
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3538 </p> |
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3539 <p xml:lang="lat"> |
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3540 <s xml:id="echoid-s284" xml:space="preserve"> |
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3541 sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi><mi>H</mi><mi>M</mi></mstyle></math>, angulus rectus <lb/> |
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3542 Maior est ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>α</mi></mstyle></math> vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>O</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>T</mi><mi>V</mi></mstyle></math>. nam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>T</mi><mi>V</mi></mstyle></math> est in eadem ratione <lb/> |
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3543 et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>H</mi></mstyle></math> maior est quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi></mstyle></math>. <lb/> |
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3544 Maior est etiam ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>δ</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>X</mi><mi>Y</mi></mstyle></math>. nam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>ɛ</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>X</mi><mi>Y</mi></mstyle></math> est eadem, |
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3545 et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>δ</mi></mstyle></math> est maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mo>\</mo><mi>v</mi><mi>a</mi><mi>r</mi><mi>e</mi><mi>s</mi><mi>p</mi><mi>s</mi><mi>i</mi><mi>l</mi><mi>o</mi><mi>n</mi></mstyle></math>. <lb/> |
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3546 Hæc ita se habent, quia <emph style="super">angulus</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>H</mi><mi>V</mi></mstyle></math> est obtusus maior scilicet recto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi><mi>H</mi><mi>M</mi></mstyle></math>. |
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3547 et alter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>H</mi><mi>X</mi></mstyle></math> est <lb/> |
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3548 minor eadem ratio. <lb/> |
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3549 Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>T</mi><mi>V</mi></mstyle></math> est minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>. <lb/> |
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3550 Ac etiam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Y</mi><mi>X</mi></mstyle></math> est minor quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>. |
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3551 <lb/>[<emph style="it">tr: |
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3552 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi><mi>H</mi><mi>M</mi></mstyle></math> be a right angle <lb/> |
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3553 The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>α</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>O</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>T</mi><mi>V</mi></mstyle></math> is greater, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>T</mi><mi>V</mi></mstyle></math> is in the same ratio |
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3554 and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>H</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi></mstyle></math>. <lb/> |
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3555 The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>δ</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>X</mi><mi>Y</mi></mstyle></math> is greater, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>ɛ</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>X</mi><mi>Y</mi></mstyle></math> is the same, |
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3556 and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>δ</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mo>\</mo><mi>v</mi><mi>a</mi><mi>r</mi><mi>e</mi><mi>s</mi><mi>p</mi><mi>s</mi><mi>i</mi><mi>l</mi><mi>o</mi><mi>n</mi></mstyle></math>. <lb/> |
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3557 These are had thus, because angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>H</mi><mi>V</mi></mstyle></math> is obtuse, clearly greater than the right angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi><mi>H</mi><mi>M</mi></mstyle></math>, |
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3558 and the other, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>H</mi><mi>X</mi></mstyle></math>, is less than the same ratio. <lb/> |
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3559 Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>T</mi><mi>V</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>. <lb/> |
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3560 And also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Y</mi><mi>X</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>P</mi></mstyle></math>. |
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3561 </emph>]<lb/> |
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3562 </s> |
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3563 </p> |
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3564 <p xml:lang="lat"> |
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3565 <s xml:id="echoid-s285" xml:space="preserve"> |
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3566 Hæc facile applicantur <lb/> |
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3567 ad diagramma <lb/> |
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3568 precedens |
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3569 <lb/>[<emph style="it">tr: |
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3570 These are easily applied to the preceding diagram. |
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3571 </emph>]<lb/> |
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3572 </s> |
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3573 </p> |
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3574 <pb file="add_6787_f230v" o="230v" n="459"/> |
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3575 <pb file="add_6787_f231" o="231" n="460"/> |
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3576 <div xml:id="echoid-div66" type="page_commentary" level="2" n="66"> |
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3577 <p> |
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3578 <s xml:id="echoid-s286" xml:space="preserve">[<emph style="it">Note: |
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3579 <p> |
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3580 <s xml:id="echoid-s286" xml:space="preserve"> |
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3581 This page gives numerical examples for Propositions III and VI from the 'Sexti dati' in Chapter 19 |
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3582 of Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). |
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3583 </s> |
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3584 <lb/> |
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3585 <quote xml:lang="lat"> |
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3586 III. <lb/> |
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3587 Data summa vel differentia duarum peripheriarum, quarum transsinuosae datam habeant rationem, dantur singulæ. |
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3588 </quote> |
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3589 <lb/> |
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3590 <quote> |
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3591 III.Given the sum or difference of two arcs, whose secants are in a given ratio, the arcs are given individually. |
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3592 </quote> |
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3593 <lb/> |
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3594 <quote xml:lang="lat"> |
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3595 VII. <lb/> |
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3596 Data summa vel differentia duarum peripheriarum, quarum prosinus datam habeant rationem, dantur singulares peripheriæ. |
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3597 </quote> |
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3598 <lb/> |
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3599 <quote> |
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3600 VII.Given the sum or difference of two arcs, whose tangents are in a given ratio, the arcs are given individually. |
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3601 </quote> |
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3602 </p> |
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3603 </emph>] |
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3604 <lb/><lb/></s></p></div> |
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3605 <head xml:id="echoid-head74" xml:space="preserve" xml:lang="lat"> |
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3606 Vieta. lib. 8. resp. <lb/> |
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3607 pag. 38. <lb/> |
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3608 <foreign xml:lang="gre">parapompe</foreign>. 3. Data summa vel differentia. <lb/> |
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3609 pag. 37.b. <lb/> |
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3610 <foreign xml:lang="gre">parapompe</foreign>. data septimi <lb/> |
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3611 <lb/>[<emph style="it">tr: |
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3612 Viète, Responsorum liber VIII, page 38v. On sines. |
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3613 </emph>]<lb/> |
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3614 </head> |
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3615 <pb file="add_6787_f231v" o="231v" n="461"/> |
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3616 <pb file="add_6787_f232" o="232" n="462"/> |
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3617 <div xml:id="echoid-div67" type="page_commentary" level="2" n="67"> |
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3618 <p> |
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3619 <s xml:id="echoid-s288" xml:space="preserve">[<emph style="it">Note: |
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3620 <p> |
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3621 <s xml:id="echoid-s288" xml:space="preserve"> |
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3622 This page gives numerical examples for Propositions I and II from the 'Sexti dati' in Chapter 19 |
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3623 of Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). |
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3624 </s> |
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3625 <lb/> |
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3626 <quote xml:lang="lat"> |
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3627 I. <lb/> |
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3628 Data peripheria composita e duabus peripheriis, quarum transsinuosae datam habeant rationem, dantur singulæ. |
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3629 </quote> |
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3630 <lb/> |
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3631 <quote> |
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3632 I.Given an arc composed of two others, whose secants are in a given ratio, each is known individually. |
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3633 </quote> |
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3634 <lb/><quote xml:lang="lat"> |
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3635 II. <lb/> |
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3636 Data differentia duarum perpheriarum, quarum transsinuosæ datam habeant rationem, dantur singulæ. |
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3637 </quote> |
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3638 <lb/> |
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3639 <quote> |
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3640 II.Given the difference of two arcs, whose secants are in a given ratio, each is given individually. |
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3641 </quote> |
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3642 </p> |
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3643 </emph>] |
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3644 <lb/><lb/></s></p></div> |
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3645 <head xml:id="echoid-head75" xml:space="preserve" xml:lang="lat"> |
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3646 Vieta. lib. 8. resp. <lb/> |
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3647 pag. 38. <lb/> |
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3648 Data peripheria <lb/> |
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3649 compositam <lb/> |
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3650 Data <emph style="st">peripheria</emph> differentia <lb/> |
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3651 duarum perpheriarum |
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3652 <lb/>[<emph style="it">tr: |
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3653 Viète, Responsorum liber VIII, page 38. <lb/> |
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3654 Given the sum of the arcs <lb/> |
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3655 Given the difference of two arcs |
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3656 </emph>]<lb/> |
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3657 </head> |
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3658 <p xml:lang="lat"> |
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3659 <s xml:id="echoid-s290" xml:space="preserve"> |
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3660 Menda in Vieta |
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3661 <lb/>[<emph style="it">tr: |
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3662 Wrong in Viète |
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3663 </emph>]<lb/> |
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3664 </s> |
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3665 </p> |
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3666 <pb file="add_6787_f232v" o="232v" n="463"/> |
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3667 <pb file="add_6787_f233" o="233" n="464"/> |
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3668 <div xml:id="echoid-div68" type="page_commentary" level="2" n="68"> |
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3669 <p> |
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3670 <s xml:id="echoid-s291" xml:space="preserve">[<emph style="it">Note: |
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3671 <p> |
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3672 <s xml:id="echoid-s291" xml:space="preserve"> |
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3673 This page gives numerical examples for Proposition VI from the 'Sexti dati' in Chapter 19 |
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3674 of Viète's <emph style="it">Variorum responsorum liber VIII</emph> (1593). |
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3675 </s> |
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3676 <quote xml:lang="lat"> |
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3677 VI. <lb/> |
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3678 Data summa vel differentia duarum perpheriarum, quarum sinus datam habeant rationem, |
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3679 dantur singulares peripheriæ. |
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3680 </quote> |
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3681 <lb/> |
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3682 <quote> |
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3683 VI.Given the sum or difference of two arcs, whose sines are in a given ratio, each arc is given individually. |
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3684 </quote> |
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3685 </p> |
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3686 </emph>] |
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3687 <lb/><lb/></s></p></div> |
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3688 <head xml:id="echoid-head76" xml:space="preserve" xml:lang="lat"> |
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3689 Vieta. resp. lib. 8. pag. 37. b. De sinubus |
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3690 <lb/>[<emph style="it">tr: |
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3691 Viète, Responsorum liber VIII, page 38v. On sines. |
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3692 </emph>]<lb/> |
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3693 </head> |
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3694 <pb file="add_6787_f233v" o="233v" n="465"/> |
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3695 <pb file="add_6787_f234" o="234" n="466"/> |
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3696 <pb file="add_6787_f234v" o="234v" n="467"/> |
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3697 <pb file="add_6787_f235" o="235" n="468"/> |
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3698 <pb file="add_6787_f235v" o="235v" n="469"/> |
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3699 <div xml:id="echoid-div69" type="page_commentary" level="2" n="69"> |
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3700 <p> |
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3701 <s xml:id="echoid-s293" xml:space="preserve">[<emph style="it">Note: |
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3702 <p> |
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3703 <s xml:id="echoid-s293" xml:space="preserve"> |
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3704 One of several pages containing a mnemonic for the digits of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>π</mi></mstyle></math>, |
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3705 by making <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>=</mo><mi>a</mi></mstyle></math>, 2 = b <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>,</mo></mstyle></math> 3 = c <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>,</mo><mi>a</mi><mi>n</mi><mi>d</mi><mi>s</mi><mi>o</mi><mi>o</mi><mi>n</mi><mo>.</mo><mi>I</mi><mi>n</mi><mi>t</mi><mi>h</mi><mi>i</mi><mi>s</mi><mi>c</mi><mi>a</mi><mi>s</mi><mi>e</mi><mi>H</mi><mi>a</mi><mi>r</mi><mi>r</mi><mi>i</mi><mi>o</mi><mi>t</mi><mi>g</mi><mi>o</mi><mi>e</mi><mi>s</mi><mi>o</mi><mi>n</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>a</mi><mi>k</mi><mi>e</mi><mi>w</mi><mi>o</mi><mi>r</mi><mi>d</mi><mi>s</mi><mi>a</mi><mi>n</mi><mi>d</mi><mi>p</mi><mi>h</mi><mi>r</mi><mi>a</mi><mi>s</mi><mi>e</mi><mi>s</mi><mi>t</mi><mi>h</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>n</mi><mi>c</mi><mi>l</mi><mi>u</mi><mi>d</mi><mi>e</mi><mi>t</mi><mi>h</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>t</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>s</mi><mi>c</mi><mi>o</mi><mi>n</mi><mi>s</mi><mi>e</mi><mi>c</mi><mi>u</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>l</mi><mi>y</mi><mo>.</mo></mstyle></math></s> |
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3706 </p> |
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3707 </emph>] |
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3708 <lb/><lb/></s></p></div> |
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3709 <p> |
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3710 <s xml:id="echoid-s295" xml:space="preserve"> |
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3711 31415926535897 ...<lb/> |
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3712 ecehigcadaeibf ... |
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3713 </s> |
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3714 </p> |
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3715 <pb file="add_6787_f236" o="236" n="470"/> |
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3716 <pb file="add_6787_f236v" o="236v" n="471"/> |
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3717 <pb file="add_6787_f237" o="237" n="472"/> |
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3718 <pb file="add_6787_f237v" o="237v" n="473"/> |
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3719 <pb file="add_6787_f238" o="238" n="474"/> |
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3720 <pb file="add_6787_f238v" o="238v" n="475"/> |
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3721 <pb file="add_6787_f239" o="239" n="476"/> |
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3722 <pb file="add_6787_f239v" o="239v" n="477"/> |
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3723 <pb file="add_6787_f240" o="240" n="478"/> |
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3724 <pb file="add_6787_f240v" o="240v" n="479"/> |
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3725 <pb file="add_6787_f241" o="241" n="480"/> |
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3726 <pb file="add_6787_f241v" o="241v" n="481"/> |
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3727 <pb file="add_6787_f242" o="242" n="482"/> |
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3728 <pb file="add_6787_f242v" o="242v" n="483"/> |
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3729 <pb file="add_6787_f243" o="243" n="484"/> |
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3730 <pb file="add_6787_f243v" o="243v" n="485"/> |
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3731 <pb file="add_6787_f244" o="244" n="486"/> |
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3732 <pb file="add_6787_f244v" o="244v" n="487"/> |
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3733 <pb file="add_6787_f245" o="245" n="488"/> |
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3734 <div xml:id="echoid-div70" type="page_commentary" level="2" n="70"> |
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3735 <p> |
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3736 <s xml:id="echoid-s296" xml:space="preserve">[<emph style="it">Note: |
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3737 <p> |
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3738 <s xml:id="echoid-s296" xml:space="preserve"> |
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3739 For the historical and mathematical context of Add MS 6787, f. 245 to f. 248 see Beery and Stedall, 2009. <lb/> |
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3740 </s> |
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|
3741 </p> |
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3742 </emph>] |
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3743 <lb/><lb/></s></p></div> |
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3744 <head xml:id="echoid-head77" xml:space="preserve"> |
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3745 Of unæquall progression <lb/> |
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3746 of sines. For S.W.L. |
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3747 </head> |
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3748 <pb file="add_6787_f245v" o="245v" n="489"/> |
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3749 <pb file="add_6787_f246" o="246" n="490"/> |
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3750 <div xml:id="echoid-div71" type="page_commentary" level="2" n="71"> |
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3751 <p> |
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3752 <s xml:id="echoid-s298" xml:space="preserve">[<emph style="it">Note: |
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3753 <p> |
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3754 <s xml:id="echoid-s298" xml:space="preserve"> |
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3755 Propositions on triangular numbers; |
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3756 this sheet was probably placed here deliberately because of the importance of triangular numbers |
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3757 in Harriot's method of interpolation (se Beery and Stedall, 2009.) |
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3758 </s> |
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3759 <lb/> |
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3760 <s xml:id="echoid-s299" xml:space="preserve"> |
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3761 The refrence to Viète is to his <emph style="it">Variotum responsorum liber VII</emph>, |
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3762 Chapter IX, Proposition 14. |
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3763 </s> |
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3764 <lb/> |
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3765 <quote xml:lang="lat"> |
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3766 Propositio XIV. <lb/> |
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3767 Si fuerint lineae quotcunque æqualiter sese excedentes, fit autem prima excessui æqualis: |
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3768 octuplum ejus quod fit sub minima & composita ex omibus, adjunctum minimae quadrato, |
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3769 æquatur quadrato compositæ ex minima & extrema dupla. |
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3770 </quote> |
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3771 <lb/> |
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3772 <quote> |
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3773 If there are any number of lines exceeding each other, and moreover the first differences are equal, |
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3774 then eight times the product of the least and the sum of all, added to the square of the least, |
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3775 is equal to the square of the sum of the least and twice the greatest. |
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3776 </quote> |
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3777 <lb/> |
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3778 <s xml:id="echoid-s300" xml:space="preserve"> |
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3779 The reference to Stevin is to his <emph style="it">L'arithmétique ... aussi l'algebre</emph> (1585). |
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3780 Pages 558–642 contain Stevin's treatment of the 'Quatriesme livre d'algebre de Diophante d'Alexandrie'. |
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3781 On page 634, Stevin has the following Theorem. |
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3782 </s> |
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3783 <lb/> |
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3784 <quote xml:lang="fre"> |
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3785 Nombre triangulaire multiplié par 8, & plus 1 faict quarré a sa racine commensurable. |
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3786 </quote> |
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3787 <lb/> |
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3788 <quote> |
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3789 A triangular number multiplied by 8, plus 1, makes a square commensurable with its root. |
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3790 </quote> |
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3791 <lb/> |
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3792 <s xml:id="echoid-s301" xml:space="preserve"> |
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3793 The reference to Maurolico is to his <emph style="it">Arithmeticorum libri duo</emph> (1575), |
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|
3794 Propostion 54 (page 24). |
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3795 </s> |
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3796 <lb/> |
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3797 <quote xml:lang="lat"> |
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3798 Omnis triangulus octuplatus cum unitate, conficit sequentis imparis quadratum. |
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3799 </quote> |
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3800 <lb/> |
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3801 <quote> |
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3802 Eight times any triangular number, plus one, makes the square of the next odd number. |
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3803 </quote> |
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3804 <lb/> |
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3805 <s xml:id="echoid-s302" xml:space="preserve"> |
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3806 As an example, Maurolico gave <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>8</mn><mo>×</mo><mn>1</mn><mn>5</mn><mo>+</mo><mn>1</mn><mo>=</mo><mn>1</mn><mn>2</mn><mn>1</mn></mstyle></math>; |
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3807 note that 15 is the fifth triangular number, 11 is the sixth odd number. |
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3808 </s> |
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3809 <lb/> |
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3810 <s xml:id="echoid-s303" xml:space="preserve"> |
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3811 There is also a reference lower down to Maurolico's Proposition 11 (page 6): |
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3812 </s> |
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3813 <lb/> |
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3814 <quote xml:lang="lat"> |
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3815 Omnis numerus triangulo deinceps: aequatur quadratis lateris trianguli maioris. |
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3816 </quote> |
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3817 <lb/> |
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3818 <quote> |
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3819 Every triangular number joined with the preceding triangular number makes the square of its side. |
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3820 </quote> |
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3821 <lb/> |
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3822 <s xml:id="echoid-s304" xml:space="preserve"> |
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3823 Maurolico's example is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>5</mn><mo>+</mo><mn>1</mn><mn>0</mn><mo>=</mo><mn>2</mn><mn>5</mn></mstyle></math>. |
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3824 </s> |
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3825 </p> |
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3826 </emph>] |
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3827 <lb/><lb/></s></p></div> |
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3828 <head xml:id="echoid-head78" xml:space="preserve" xml:lang="lat"> |
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|
3829 Propositio |
|
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3830 <lb/>[<emph style="it">tr: |
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|
3831 Proposition |
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3832 </emph>]<lb/> |
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|
3833 </head> |
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3834 <p xml:lang="lat"> |
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3835 <s xml:id="echoid-s306" xml:space="preserve"> |
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|
3836 plutarchus platonica <lb/> |
|
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|
3837 quæstione 4<emph style="super">a</emph>. |
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3838 <lb/>[<emph style="it">tr: |
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|
3839 Plutarch on Plato, question 4. |
|
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3840 </emph>]<lb/> |
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|
3841 </s> |
|
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3842 <lb/> |
|
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3843 <s xml:id="echoid-s307" xml:space="preserve"> |
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|
3844 Vieta resp. pa. 15. |
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3845 <lb/>[<emph style="it">tr: |
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|
3846 Viete, <emph style="it">Liber variorum responsorum</emph>, page 15. |
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|
3847 </emph>]<lb/> |
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|
3848 </s> |
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3849 <lb/> |
|
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3850 <s xml:id="echoid-s308" xml:space="preserve"> |
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|
3851 Maurolicus pr. 54. arith. |
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3852 <lb/>[<emph style="it">tr: |
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|
3853 Maurolico, <emph style="it">Arithmetica</emph>, Proposition 54. |
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|
3854 </emph>]<lb/> |
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|
3855 </s> |
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|
3856 <lb/> |
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|
3857 <s xml:id="echoid-s309" xml:space="preserve"> |
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|
3858 Stevin . pag. 634. arith. <lb/> |
|
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|
3859 in diophanto. |
|
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3860 <lb/>[<emph style="it">tr: |
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|
3861 Stevin, page 634, <emph style="it">Arithmetic</emph>, on Diophantus. |
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|
3862 </emph>]<lb/> |
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|
3863 </s> |
|
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|
3864 </p> |
|
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|
3865 <p xml:lang="lat"> |
|
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|
3866 <s xml:id="echoid-s310" xml:space="preserve"> |
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|
3867 Trianguli numeri octuplum, plus quadrato unitatis: |
|
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|
3868 æqualie est quadrato <lb/> |
|
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|
3869 facto a duplo latere trianguli plus unitate. |
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|
3870 <lb/>[<emph style="it">tr: |
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|
3871 Eight times a trinagular number plus the square of one |
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|
3872 is equal to the square of double the side of the triangular number, plus one. |
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|
3873 </emph>]<lb/> |
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|
3874 </s> |
|
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|
3875 </p> |
|
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|
3876 <p xml:lang="lat"> |
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|
3877 <s xml:id="echoid-s311" xml:space="preserve"> |
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|
3878 Sit latus trianguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>. |
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|
3879 <lb/>[<emph style="it">tr: |
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|
3880 Let the side of the triangle be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>. |
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|
3881 </emph>]<lb/> |
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|
3882 </s> |
|
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|
3883 <lb/> |
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|
3884 <s xml:id="echoid-s312" xml:space="preserve"> |
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|
3885 Tum triangulus erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo maxsize="1">)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> |
|
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|
3886 <lb/>[<emph style="it">tr: |
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|
3887 Then the triangular number will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo maxsize="1">)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> |
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|
3888 </emph>]<lb/> |
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|
3889 </s> |
|
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|
3890 <lb/> |
|
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|
3891 <s xml:id="echoid-s313" xml:space="preserve"> |
|
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|
3892 Unde propositio cum demonstratione in notis logisticis ita se habet: |
|
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|
3893 <lb/>[<emph style="it">tr: |
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|
3894 Whence we have the proposition with its demonstration in arithmetic notation thus: |
|
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|
3895 </emph>]<lb/> |
|
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|
3896 </s> |
|
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|
3897 </p> |
|
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|
3898 <head xml:id="echoid-head79" xml:space="preserve" xml:lang="lat"> |
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|
3899 Propositio |
|
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|
3900 <lb/>[<emph style="it">tr: |
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|
3901 Proposition |
|
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|
3902 </emph>]<lb/> |
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|
3903 </head> |
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|
3904 <p xml:lang="lat"> |
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3905 <s xml:id="echoid-s314" xml:space="preserve"> |
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|
3906 Marol. pr. 11 |
|
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|
3907 <lb/>[<emph style="it">tr: |
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|
3908 Maurolico, Proposition 11 |
|
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|
3909 </emph>]<lb/> |
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|
3910 </s> |
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|
3911 <lb/> |
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3912 <s xml:id="echoid-s315" xml:space="preserve"> |
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|
3913 Omnis numerus triangulus, plus triangulo deinceps <emph style="st">priori</emph>: æquatur <lb/> |
|
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|
3914 quadrato lateris trianguli maioris. |
|
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|
3915 <lb/>[<emph style="it">tr: |
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3916 Every triangular number, plus the triangular numebr following, |
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3917 is equal to the square of the side of the greater triangular number. |
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3918 </emph>]<lb/> |
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3919 </s> |
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3920 <lb/> |
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3921 <s xml:id="echoid-s316" xml:space="preserve"> |
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3922 Sit latus maioris trianguli. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>. |
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3923 <lb/>[<emph style="it">tr: |
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3924 Let the side of the greater triangular number be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi></mstyle></math>. |
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3925 </emph>]<lb/> |
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3926 </s> |
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3927 <lb/> |
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3928 <s xml:id="echoid-s317" xml:space="preserve"> |
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3929 latus minoris triang: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>-</mo><mn>1</mn></mstyle></math>. |
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3930 <lb/>[<emph style="it">tr: |
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3931 the side of the smaller triangular number is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>-</mo><mn>1</mn></mstyle></math>. |
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3932 </emph>]<lb/> |
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3933 </s> |
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3934 <lb/> |
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3935 <s xml:id="echoid-s318" xml:space="preserve"> |
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3936 Ergo: maior triangulus. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo maxsize="1">)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>. |
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3937 <lb/>[<emph style="it">tr: |
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3938 Therefore the greater triangular number is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>n</mi><mo maxsize="1">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo maxsize="1">)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>. |
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3939 </emph>]<lb/> |
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3940 </s> |
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3941 <lb/> |
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3942 <s xml:id="echoid-s319" xml:space="preserve"> |
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3943 Minor triangulus. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mo maxsize="1">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>. |
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3944 <lb/>[<emph style="it">tr: |
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3945 the smaller triangular number is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mo maxsize="1">(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>. |
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3946 </emph>]<lb/> |
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3947 </s> |
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3948 <lb/> |
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3949 <s xml:id="echoid-s320" xml:space="preserve"> |
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|
3950 Unde propositio cum demonstration in notis logisticis ita se habet: |
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3951 <lb/>[<emph style="it">tr: |
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3952 Whence we have the proposition with its demonstration in arithmetic notation thus: |
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3953 </emph>]<lb/> |
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3954 </s> |
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3955 </p> |
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3956 <pb file="add_6787_f246v" o="246v" n="491"/> |
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3957 <pb file="add_6787_f247" o="247" n="492"/> |
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3958 <head xml:id="echoid-head80" xml:space="preserve" xml:lang="lat"> |
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|
3959 Problema |
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3960 <lb/>[<emph style="it">tr: |
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|
3961 Problem |
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3962 </emph>]<lb/> |
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3963 </head> |
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3964 <p> |
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3965 <s xml:id="echoid-s321" xml:space="preserve"> |
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3966 A progression increasing being given, whose |
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3967 <emph style="super"><emph style="st">next</emph> first progresionall </emph> |
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3968 <emph style="st">first</emph> differences <lb/> |
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3969 are unæquall, & the <emph style="st">second</emph> |
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|
3970 <emph style="super">next <emph style="st">after</emph> differences</emph> æquall: |
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|
3971 to devide the sayde <lb/> |
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|
3972 progression into a fewer nomber of progresionall partes. |
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|
3973 </s> |
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3974 <lb/> |
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3975 <s xml:id="echoid-s322" xml:space="preserve"> |
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|
3976 The first case. as of the <emph style="super">progresionall</emph> differences decreasing, <lb/> |
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|
3977 as it is in the progression of sines. |
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|
3978 </s> |
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|
3979 </p> |
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|
3980 <p> |
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|
3981 <s xml:id="echoid-s323" xml:space="preserve"> |
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|
3982 Example of the progresion <emph style="super">given</emph>. |
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|
3983 </s> |
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|
3984 </p> |
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|
3985 <p> |
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|
3986 <s xml:id="echoid-s324" xml:space="preserve"> |
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|
3987 <sc> |
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|
3988 'Arkes' are arcs of a fixed circle, that is, measures of angle. |
|
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|
3989 Each sine (or number) corresponds to, or is 'answerable to', an arc (or angle). |
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|
3990 (Hence the modern terminology 'arcsin'.) |
|
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|
3991 </sc> |
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|
3992 This example I have <lb/> |
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|
3993 so set downe as though the <lb/> |
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|
3994 nombers were answerable <lb/> |
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|
3995 to arkes. |
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|
3996 </s> |
|
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|
3997 <s xml:id="echoid-s325" xml:space="preserve"> |
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|
3998 That by it you may <lb/> |
|
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|
3999 se the use of the problem <lb/> |
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|
4000 for sines. |
|
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4001 </s> |
|
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4002 </p> |
|
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4003 <p> |
|
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4004 <s xml:id="echoid-s326" xml:space="preserve"> |
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|
4005 Suppose that it be required to find the nomber answerable to 37''. |
|
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4006 </s> |
|
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|
4007 <lb/> |
|
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4008 <s xml:id="echoid-s327" xml:space="preserve"> |
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|
4009 The number answerable <emph style="super">to</emph> 30'' is 2280. that which is answerable <lb/> |
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|
4010 to 40'' is 2925. there difference is 645. |
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|
4011 </s> |
|
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4012 <s xml:id="echoid-s328" xml:space="preserve"> |
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|
4013 And because the other <lb/> |
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4014 differences of the same <emph style="super">order</emph> ranke are unæquall, the rule of pro-<lb/> |
|
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|
4015 portion will not find the number desired. But it must be <lb/> |
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|
4016 found by a speciall Canon. <emph style="st">which followeth</emph> |
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|
4017 </s> |
|
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4018 <s xml:id="echoid-s329" xml:space="preserve"> |
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|
4019 The purpose <lb/> |
|
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4020 of <emph style="st">which canon</emph> <emph style="super">thereof</emph> must be |
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|
4021 <emph style="st">for to find</emph> (because 30''& 40'' do differ <lb/> |
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|
4022 [by] 10''.) to find the <emph style="st"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mstyle></math></emph> |
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4023 <emph style="super">first tenth</emph> parte progresionall, that is to say that number. |
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|
4024 </s> |
|
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|
4025 </p> |
|
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|
4026 <pb file="add_6787_f247v" o="247v" n="493"/> |
|
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|
4027 <pb file="add_6787_f248" o="248" n="494"/> |
|
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|
4028 <div xml:id="echoid-div72" type="page_commentary" level="2" n="72"> |
|
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|
4029 <p> |
|
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|
4030 <s xml:id="echoid-s330" xml:space="preserve">[<emph style="it">Note: |
|
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|
4031 <p> |
|
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|
4032 <s xml:id="echoid-s330" xml:space="preserve"> |
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|
4033 The table of 'pretend' sines from Add MS 6787, f. 247, now interpolated to give sines for every minute |
|
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|
4034 from 30minutes to 50 minutes. |
|
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|
4035 </s> |
|
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|
4036 </p> |
|
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|
4037 </emph>] |
|
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|
4038 <lb/><lb/></s></p></div> |
|
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|
4039 <pb file="add_6787_f248v" o="248v" n="495"/> |
|
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|
4040 <pb file="add_6787_f249" o="249" n="496"/> |
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|
4041 <pb file="add_6787_f249v" o="249v" n="497"/> |
|
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|
4042 <pb file="add_6787_f250" o="250" n="498"/> |
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|
4043 <pb file="add_6787_f250v" o="250v" n="499"/> |
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|
4044 <p> |
|
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|
4045 <s xml:id="echoid-s332" xml:space="preserve"> |
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|
4046 Serjeant Harris. <lb/> |
|
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|
4047 Mr Martin. <lb/> |
|
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|
4048 Mr Waldon. <lb/> |
|
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|
4049 Mr More. <lb/> |
|
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|
4050 Mr Karbile <lb/> |
|
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|
4051 Mr Walters. <lb/> |
|
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|
4052 Mr Dorrel |
|
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|
4053 </s> |
|
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|
4054 </p> |
|
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|
4055 <p> |
|
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|
4056 <s xml:id="echoid-s333" xml:space="preserve"> |
|
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|
4057 April 26. ho. 7. <lb/> |
|
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|
4058 first subpena <lb/> |
|
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|
4059 George Sanderson's man |
|
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|
4060 </s> |
|
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|
4061 </p> |
|
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|
4062 <p> |
|
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|
4063 <s xml:id="echoid-s334" xml:space="preserve"> |
|
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|
4064 Mr Nicholas Lower <lb/> |
|
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|
4065 at a dues mans home <emph style="super">[???]</emph> <lb/> |
|
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|
4066 within the meremayde <lb/> |
|
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|
4067 in cheapside. |
|
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|
4068 </s> |
|
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|
4069 </p> |
|
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|
4070 <p> |
|
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|
4071 <s xml:id="echoid-s335" xml:space="preserve"> |
|
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|
4072 Bish. of Elyes booke. <lb/> |
|
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|
4073 Wright. <lb/> |
|
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|
4074 Thesaro politico. 2. <lb/> |
|
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|
4075 Onuphrius in platinam. |
|
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|
4076 </s> |
|
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|
4077 </p> |
|
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|
4078 <p> |
|
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|
4079 <s xml:id="echoid-s336" xml:space="preserve"> |
|
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|
4080 [???] <lb/> |
|
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|
4081 Mappes <lb/> |
|
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|
4082 globes |
|
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|
4083 </s> |
|
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|
4084 </p> |
|
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|
4085 <pb file="add_6787_f251" o="251" n="500"/> |
|
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|
4086 <pb file="add_6787_f251v" o="251v" n="501"/> |
|
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|
4087 <pb file="add_6787_f252" o="252" n="502"/> |
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4088 <pb file="add_6787_f252v" o="252v" n="503"/> |
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4089 <div xml:id="echoid-div73" type="page_commentary" level="2" n="73"> |
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4090 <p> |
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4091 <s xml:id="echoid-s337" xml:space="preserve">[<emph style="it">Note: |
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4092 <p> |
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4093 <s xml:id="echoid-s337" xml:space="preserve"> |
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|
4094 A continuation of the calculations on Add MS 6787, f. 56, now extended to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>v</mi><mn>5</mn></msup></mrow></mstyle></math>. <lb/> |
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4095 On this page Harriot also changes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>+</mo><mn>1</mn></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mo>+</mo><mi>b</mi></mstyle></math>. |
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4096 </s> |
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4097 </p> |
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4098 </emph>] |
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4099 <lb/><lb/></s></p></div> |
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4100 <pb file="add_6787_f253" o="253" n="504"/> |
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4101 <pb file="add_6787_f253v" o="253v" n="505"/> |
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4102 <pb file="add_6787_f254" o="254" n="506"/> |
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|
4103 <pb file="add_6787_f254v" o="254v" n="507"/> |
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4104 <pb file="add_6787_f255" o="255" n="508"/> |
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4105 <pb file="add_6787_f255v" o="255v" n="509"/> |
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4106 <pb file="add_6787_f256" o="256" n="510"/> |
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4107 <pb file="add_6787_f256v" o="256v" n="511"/> |
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4108 <pb file="add_6787_f257" o="257" n="512"/> |
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4109 <pb file="add_6787_f257v" o="257v" n="513"/> |
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4110 <pb file="add_6787_f258" o="258" n="514"/> |
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4111 <pb file="add_6787_f258v" o="258v" n="515"/> |
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4112 <pb file="add_6787_f259" o="259" n="516"/> |
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4113 <pb file="add_6787_f259v" o="259v" n="517"/> |
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4114 <pb file="add_6787_f260" o="260" n="518"/> |
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4115 <pb file="add_6787_f260v" o="260v" n="519"/> |
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4116 <pb file="add_6787_f261" o="261" n="520"/> |
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4117 <pb file="add_6787_f261v" o="261v" n="521"/> |
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4118 <pb file="add_6787_f262" o="262" n="522"/> |
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4119 <pb file="add_6787_f262v" o="262v" n="523"/> |
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4120 <pb file="add_6787_f263" o="263" n="524"/> |
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4121 <pb file="add_6787_f263v" o="263v" n="525"/> |
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|
4122 <pb file="add_6787_f264" o="264" n="526"/> |
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|
4123 <pb file="add_6787_f264v" o="264v" n="527"/> |
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4124 <pb file="add_6787_f265" o="265" n="528"/> |
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22d6a63640c6
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|
4125 <pb file="add_6787_f265v" o="265v" n="529"/> |
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22d6a63640c6
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|
4126 <pb file="add_6787_f266" o="266" n="530"/> |
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22d6a63640c6
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4127 <pb file="add_6787_f266v" o="266v" n="531"/> |
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22d6a63640c6
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4128 <pb file="add_6787_f267" o="267" n="532"/> |
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4129 <pb file="add_6787_f267v" o="267v" n="533"/> |
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22d6a63640c6
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4130 <pb file="add_6787_f268" o="268" n="534"/> |
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22d6a63640c6
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4131 <pb file="add_6787_f268v" o="268v" n="535"/> |
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22d6a63640c6
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4132 <pb file="add_6787_f269" o="269" n="536"/> |
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22d6a63640c6
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|
4133 <pb file="add_6787_f269v" o="269v" n="537"/> |
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4134 <pb file="add_6787_f270" o="270" n="538"/> |
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4135 <pb file="add_6787_f270v" o="270v" n="539"/> |
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22d6a63640c6
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4136 <pb file="add_6787_f271" o="271" n="540"/> |
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22d6a63640c6
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4137 <pb file="add_6787_f271v" o="271v" n="541"/> |
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22d6a63640c6
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4138 <pb file="add_6787_f272" o="272" n="542"/> |
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4139 <pb file="add_6787_f272v" o="272v" n="543"/> |
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4140 <pb file="add_6787_f273" o="273" n="544"/> |
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4141 <pb file="add_6787_f273v" o="273v" n="545"/> |
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4142 <pb file="add_6787_f274" o="274" n="546"/> |
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22d6a63640c6
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|
4143 <p> |
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4144 <s xml:id="echoid-s339" xml:space="preserve"> |
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|
4145 Cloth for a [???] <lb/> |
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|
4146 4 yardes of [???] <lb/> |
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|
4147 Velvet <lb/> |
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|
4148 [???] for doublet and hose <lb/> |
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|
4149 & taffeta <lb/> |
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|
4150 4 yards |
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|
4151 </s> |
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|
4152 </p> |
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|
4153 <pb file="add_6787_f274v" o="274v" n="547"/> |
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|
4154 <pb file="add_6787_f275" o="275" n="548"/> |
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4155 <pb file="add_6787_f275v" o="275v" n="549"/> |
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4156 <pb file="add_6787_f276" o="276" n="550"/> |
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4157 <pb file="add_6787_f276v" o="276v" n="551"/> |
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4158 <pb file="add_6787_f277" o="277" n="552"/> |
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4159 <pb file="add_6787_f277v" o="277v" n="553"/> |
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4160 <pb file="add_6787_f278" o="278" n="554"/> |
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4161 <pb file="add_6787_f278v" o="278v" n="555"/> |
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4162 <pb file="add_6787_f279" o="279" n="556"/> |
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4163 <pb file="add_6787_f279v" o="279v" n="557"/> |
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4164 <pb file="add_6787_f280" o="280" n="558"/> |
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4165 <pb file="add_6787_f280v" o="280v" n="559"/> |
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4166 <pb file="add_6787_f281" o="281" n="560"/> |
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4167 <pb file="add_6787_f281v" o="281v" n="561"/> |
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4168 <pb file="add_6787_f282" o="282" n="562"/> |
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4169 <pb file="add_6787_f282v" o="282v" n="563"/> |
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4170 <pb file="add_6787_f283" o="283" n="564"/> |
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4171 <pb file="add_6787_f283v" o="283v" n="565"/> |
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4172 <pb file="add_6787_f284" o="284" n="566"/> |
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4173 <pb file="add_6787_f284v" o="284v" n="567"/> |
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4174 <pb file="add_6787_f285" o="285" n="568"/> |
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4175 <pb file="add_6787_f285v" o="285v" n="569"/> |
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4176 <pb file="add_6787_f286" o="286" n="570"/> |
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4177 <pb file="add_6787_f286v" o="286v" n="571"/> |
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4178 <pb file="add_6787_f287" o="287" n="572"/> |
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4179 <pb file="add_6787_f287v" o="287v" n="573"/> |
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4180 <pb file="add_6787_f288" o="288" n="574"/> |
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4181 <pb file="add_6787_f288v" o="288v" n="575"/> |
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4182 <pb file="add_6787_f289" o="289" n="576"/> |
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4183 <pb file="add_6787_f289v" o="289v" n="577"/> |
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4184 <pb file="add_6787_f290" o="290" n="578"/> |
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4185 <pb file="add_6787_f290v" o="290v" n="579"/> |
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4186 <pb file="add_6787_f291" o="291" n="580"/> |
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4187 <pb file="add_6787_f291v" o="291v" n="581"/> |
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4188 <pb file="add_6787_f292" o="292" n="582"/> |
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4189 <pb file="add_6787_f292v" o="292v" n="583"/> |
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4190 <pb file="add_6787_f293" o="293" n="584"/> |
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4191 <pb file="add_6787_f293v" o="293v" n="585"/> |
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4192 <pb file="add_6787_f294" o="294" n="586"/> |
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4193 <pb file="add_6787_f294v" o="294v" n="587"/> |
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4194 <pb file="add_6787_f295" o="295" n="588"/> |
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4195 <pb file="add_6787_f295v" o="295v" n="589"/> |
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4196 <pb file="add_6787_f296" o="296" n="590"/> |
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4197 <pb file="add_6787_f296v" o="296v" n="591"/> |
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4198 <pb file="add_6787_f297" o="297" n="592"/> |
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4199 <pb file="add_6787_f297v" o="297v" n="593"/> |
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4200 <pb file="add_6787_f298" o="298" n="594"/> |
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4201 <pb file="add_6787_f298v" o="298v" n="595"/> |
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4202 <div xml:id="echoid-div74" type="page_commentary" level="2" n="74"> |
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4203 <p> |
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4204 <s xml:id="echoid-s340" xml:space="preserve">[<emph style="it">Note: |
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4205 <p> |
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4206 <s xml:id="echoid-s340" xml:space="preserve"> |
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4207 The reference on this page is to Viète, |
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4208 <emph style="it">Adrianus Romanus responsum</emph> (1595). |
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4209 </s> |
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4210 </p> |
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4211 </emph>] |
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4212 <lb/><lb/></s></p></div> |
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4213 <head xml:id="echoid-head81" xml:space="preserve" xml:lang="lat"> |
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4214 Ad Auctorium Vietæ in responso ad Romanum |
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4215 <lb/>[<emph style="it">tr: |
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4216 On teh authority of Viète in his response to Romanus |
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4217 </emph>]<lb/> |
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4218 </head> |
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4219 <p xml:lang="lat"> |
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4220 <s xml:id="echoid-s342" xml:space="preserve"> |
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4221 occasio doctrinæ <lb/> |
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4222 in his chartis |
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4223 <lb/>[<emph style="it">tr: |
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4224 the occasion for the teaching in these sheets |
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4225 </emph>]<lb/> |
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4226 </s> |
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4227 </p> |
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4228 <pb file="add_6787_f299" o="299" n="596"/> |
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4229 <pb file="add_6787_f299v" o="299v" n="597"/> |
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4230 <pb file="add_6787_f300" o="300" n="598"/> |
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4231 <pb file="add_6787_f300v" o="300v" n="599"/> |
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4232 <pb file="add_6787_f301" o="301" n="600"/> |
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4233 <pb file="add_6787_f301v" o="301v" n="601"/> |
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4234 <pb file="add_6787_f302" o="302" n="602"/> |
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4235 <pb file="add_6787_f302v" o="302v" n="603"/> |
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4236 <pb file="add_6787_f303" o="303" n="604"/> |
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4237 <pb file="add_6787_f303v" o="303v" n="605"/> |
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4238 <pb file="add_6787_f304" o="304" n="606"/> |
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4239 <div xml:id="echoid-div75" type="page_commentary" level="2" n="75"> |
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4240 <p> |
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|
4241 <s xml:id="echoid-s343" xml:space="preserve">[<emph style="it">Note: |
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|
4242 <p> |
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4243 <s xml:id="echoid-s343" xml:space="preserve"> |
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|
4244 Here Harriot creates further Pythagorean triples, not necessarily all integers. <lb/> |
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4245 In the first few lines, for example, he multiplies 21 by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>5</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>4</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> |
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|
4246 to obtain (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn><mn>0</mn><mn>5</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>6</mn><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>8</mn><mn>4</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>) or |
|
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4247 (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><mn>2</mn><mn>6</mn><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><mn>1</mn><mn>5</mn><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>1</mn></mstyle></math>). |
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4248 </s> |
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4249 </p> |
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4250 </emph>] |
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4251 <lb/><lb/></s></p></div> |
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4252 <pb file="add_6787_f304v" o="304v" n="607"/> |
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4253 <pb file="add_6787_f305" o="305" n="608"/> |
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4254 <div xml:id="echoid-div76" type="page_commentary" level="2" n="76"> |
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4255 <p> |
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4256 <s xml:id="echoid-s345" xml:space="preserve">[<emph style="it">Note: |
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4257 <p> |
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4258 <s xml:id="echoid-s345" xml:space="preserve"> |
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4259 This is the cover sheet of eight pages copied from Book 3, Propositions 1 and 2, |
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4260 of the <emph style="it">Sphaerica Menelai</emph> of Menelaus of Alexandria, |
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4261 as translated by Francesco Maurolico (1588). <lb/> |
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4262 The text includes a supplement by Thabit ibn Qurra (826–901). |
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4263 </s> |
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4264 </p> |
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4265 </emph>] |
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4266 <lb/><lb/></s></p></div> |
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4267 <head xml:id="echoid-head82" xml:space="preserve" xml:lang="lat"> |
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4268 Propositiones quædam <lb/> |
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4269 ex Menelauo <lb/> |
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4270 et <lb/> |
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4271 Thebit |
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4272 <lb/>[<emph style="it">tr: |
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4273 Certain propositions from Menelaus and Thebit |
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4274 </emph>]<lb/> |
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4275 </head> |
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4276 <pb file="add_6787_f305v" o="305v" n="609"/> |
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4277 <pb file="add_6787_f306" o="306" n="610"/> |
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4278 <div xml:id="echoid-div77" type="page_commentary" level="2" n="77"> |
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4279 <p> |
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4280 <s xml:id="echoid-s347" xml:space="preserve">[<emph style="it">Note: |
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4281 <p> |
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4282 <s xml:id="echoid-s347" xml:space="preserve"> |
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4283 Text from pages 36v–37v of <emph style="it">Sphaerica Menelai</emph>. |
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4284 </s> |
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4285 </p> |
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4286 </emph>] |
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4287 <lb/><lb/></s></p></div> |
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4288 <head xml:id="echoid-head83" xml:space="preserve" xml:lang="lat"> |
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4289 Ex libro 3. Spæricorum Menelai <lb/> |
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4290 secundum traditionem Maurolyci <lb/> |
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4291 1 |
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4292 <lb/>[<emph style="it">tr: |
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4293 From Book 3 of Sphærica Menelai, as translated by Maurolicus |
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4294 </emph>]<lb/> |
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4295 </head> |
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4296 <pb file="add_6787_f306v" o="306v" n="611"/> |
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4297 <div xml:id="echoid-div78" type="page_commentary" level="2" n="78"> |
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4298 <p> |
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4299 <s xml:id="echoid-s349" xml:space="preserve">[<emph style="it">Note: |
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4300 <p> |
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4301 <s xml:id="echoid-s349" xml:space="preserve"> |
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4302 Text from pages 37v–38 of <emph style="it">Sphaerica Menelai</emph>. <lb/> |
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4303 The supplement by Thabit ibn Qurra begins here and continues to f. 308. |
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4304 </s> |
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4305 </p> |
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4306 </emph>] |
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4307 <lb/><lb/></s></p></div> |
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4308 <p xml:lang="lat"> |
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4309 <s xml:id="echoid-s351" xml:space="preserve"> |
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4310 <lb/>[...]<lb/> |
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4311 </s> |
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4312 <lb/> |
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4313 <s xml:id="echoid-s352" xml:space="preserve"> |
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|
4314 Suppementum Thebitij |
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4315 <lb/>[<emph style="it">tr: |
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4316 Supplement by Thebit |
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4317 </emph>]<lb/> |
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4318 </s> |
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4319 <lb/> |
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4320 <s xml:id="echoid-s353" xml:space="preserve"> |
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4321 <lb/>[...]<lb/> |
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4322 </s> |
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4323 </p> |
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4324 <pb file="add_6787_f307" o="307" n="612"/> |
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4325 <div xml:id="echoid-div79" type="page_commentary" level="2" n="79"> |
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4326 <p> |
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4327 <s xml:id="echoid-s354" xml:space="preserve">[<emph style="it">Note: |
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4328 <p> |
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4329 <s xml:id="echoid-s354" xml:space="preserve"> |
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|
4330 Text from pages 38 from <emph style="it">Sphaerica Menelai</emph>. <lb/> |
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4331 For further work by Harriot on the 18 cases of Lemma 4, see Ad MS 6786, f. 440. |
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4332 </s> |
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4333 </p> |
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4334 </emph>] |
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4335 <lb/><lb/></s></p></div> |
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4336 <head xml:id="echoid-head84" xml:space="preserve"> |
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4337 2 |
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4338 </head> |
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4339 <pb file="add_6787_f307v" o="307v" n="613"/> |
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4340 <div xml:id="echoid-div80" type="page_commentary" level="2" n="80"> |
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4341 <p> |
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4342 <s xml:id="echoid-s356" xml:space="preserve">[<emph style="it">Note: |
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4343 <p> |
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4344 <s xml:id="echoid-s356" xml:space="preserve"> |
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|
4345 Text from page 38v of <emph style="it">Sphaerica Menelai</emph>. |
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4346 </s> |
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4347 </p> |
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4348 </emph>] |
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4349 <lb/><lb/></s></p></div> |
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4350 <pb file="add_6787_f308" o="308" n="614"/> |
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4351 <div xml:id="echoid-div81" type="page_commentary" level="2" n="81"> |
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4352 <p> |
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4353 <s xml:id="echoid-s358" xml:space="preserve">[<emph style="it">Note: |
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4354 <p> |
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4355 <s xml:id="echoid-s358" xml:space="preserve"> |
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4356 Text from pages 38v–39 of <emph style="it">Sphaerica Menelai</emph>. |
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4357 </s> |
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4358 </p> |
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4359 </emph>] |
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4360 <lb/><lb/></s></p></div> |
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4361 <head xml:id="echoid-head85" xml:space="preserve"> |
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4362 3 |
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4363 </head> |
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4364 <pb file="add_6787_f308v" o="308v" n="615"/> |
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4365 <div xml:id="echoid-div82" type="page_commentary" level="2" n="82"> |
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4366 <p> |
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4367 <s xml:id="echoid-s360" xml:space="preserve">[<emph style="it">Note: |
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4368 <p> |
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4369 <s xml:id="echoid-s360" xml:space="preserve"> |
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|
4370 Text from pages 39 of <emph style="it">Sphaerica Menelai</emph>. |
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4371 </s> |
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4372 </p> |
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4373 </emph>] |
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4374 <lb/><lb/></s></p></div> |
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|
4375 <pb file="add_6787_f309" o="309" n="616"/> |
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|
4376 <div xml:id="echoid-div83" type="page_commentary" level="2" n="83"> |
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|
4377 <p> |
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4378 <s xml:id="echoid-s362" xml:space="preserve">[<emph style="it">Note: |
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4379 <p> |
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|
4380 <s xml:id="echoid-s362" xml:space="preserve"> |
|
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|
4381 Text from pages 39–39v of <emph style="it">Sphaerica Menelai</emph>. |
|
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|
4382 </s> |
|
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|
4383 </p> |
|
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|
4384 </emph>] |
|
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|
4385 <lb/><lb/></s></p></div> |
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|
4386 <head xml:id="echoid-head86" xml:space="preserve"> |
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|
4387 4 |
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|
4388 </head> |
|
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|
4389 <pb file="add_6787_f309v" o="309v" n="617"/> |
|
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|
4390 <div xml:id="echoid-div84" type="page_commentary" level="2" n="84"> |
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|
4391 <p> |
|
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|
4392 <s xml:id="echoid-s364" xml:space="preserve">[<emph style="it">Note: |
|
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|
4393 <p> |
|
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|
4394 <s xml:id="echoid-s364" xml:space="preserve"> |
|
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|
4395 Text from pages 39v–40 of <emph style="it">Sphaerica Menelai</emph>.<lb/> |
|
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|
4396 Thabit ibn Qurra is mentioned in the final paragraph; see also Add MS 6787, f. 306v. |
|
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|
4397 </s> |
|
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|
4398 </p> |
|
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|
4399 </emph>] |
|
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|
4400 <lb/><lb/></s></p></div> |
|
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|
4401 <p xml:lang="lat"> |
|
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|
4402 <s xml:id="echoid-s366" xml:space="preserve"> |
|
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|
4403 Liber impressus fuit <lb/> |
|
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|
4404 Messanæ in freto Siculo <lb/> |
|
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|
4405 1558 |
|
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|
4406 <lb/>[<emph style="it">tr: |
|
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|
4407 The book was printed in Messina, in the Straits of Sicily, 1558. |
|
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|
4408 </emph>]<lb/> |
|
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|
4409 </s> |
|
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|
4410 </p> |
|
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|
4411 <pb file="add_6787_f310" o="310" n="618"/> |
|
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|
4412 <pb file="add_6787_f310v" o="310v" n="619"/> |
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|
4413 <pb file="add_6787_f311" o="311" n="620"/> |
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|
4414 <pb file="add_6787_f311v" o="311v" n="621"/> |
|
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|
4415 <pb file="add_6787_f312" o="312" n="622"/> |
|
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|
4416 <pb file="add_6787_f312v" o="312v" n="623"/> |
|
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|
4417 <pb file="add_6787_f313" o="313" n="624"/> |
|
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|
4418 <pb file="add_6787_f313v" o="313v" n="625"/> |
|
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|
4419 <pb file="add_6787_f314" o="314" n="626"/> |
|
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|
4420 <pb file="add_6787_f314v" o="314v" n="627"/> |
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|
4421 <pb file="add_6787_f315" o="315" n="628"/> |
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|
4422 <pb file="add_6787_f315v" o="315v" n="629"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4423 <pb file="add_6787_f316" o="316" n="630"/> |
|
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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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|
4424 <pb file="add_6787_f316v" o="316v" n="631"/> |
|
22d6a63640c6
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|
4425 <pb file="add_6787_f317" o="317" n="632"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4426 <div xml:id="echoid-div85" type="page_commentary" level="2" n="85"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4427 <p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4428 <s xml:id="echoid-s367" xml:space="preserve">[<emph style="it">Note: |
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22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4429 <p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4430 <s xml:id="echoid-s367" 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
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|
4431 One of several pages containing a mnemonic for the digits of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>π</mi></mstyle></math>, |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4432 by making <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>=</mo><mi>a</mi></mstyle></math>, 2 = b <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>,</mo></mstyle></math> 3 = c <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>,</mo><mi>a</mi><mi>n</mi><mi>d</mi><mi>s</mi><mi>o</mi><mi>o</mi><mi>n</mi><mo>.</mo></mstyle></math></s> |
|
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|
4433 </p> |
|
22d6a63640c6
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|
4434 </emph>] |
|
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|
4435 <lb/><lb/></s></p></div> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4436 <p> |
|
22d6a63640c6
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|
4437 <s xml:id="echoid-s369" 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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|
4438 cadaeibf <lb/> |
|
22d6a63640c6
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|
4439 31415926 |
|
22d6a63640c6
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|
4440 </s> |
|
22d6a63640c6
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|
4441 </p> |
|
22d6a63640c6
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|
4442 <pb file="add_6787_f317v" o="317v" n="633"/> |
|
22d6a63640c6
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|
4443 <pb file="add_6787_f318" o="318" n="634"/> |
|
22d6a63640c6
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|
4444 <pb file="add_6787_f318v" o="318v" n="635"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4445 <div xml:id="echoid-div86" type="page_commentary" level="2" n="86"> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4446 <p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4447 <s xml:id="echoid-s370" 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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|
4448 <p> |
|
22d6a63640c6
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|
4449 <s xml:id="echoid-s370" 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
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|
4450 One of several pages containing a mnemonic for the digits of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>π</mi></mstyle></math>, |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4451 by making <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mo>=</mo><mi>a</mi></mstyle></math>, 2 = b <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>,</mo></mstyle></math> 3 = c <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>,</mo><mi>a</mi><mi>n</mi><mi>d</mi><mi>s</mi><mi>o</mi><mi>o</mi><mi>n</mi><mo>.</mo></mstyle></math></s> |
|
22d6a63640c6
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|
4452 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4453 </emph>] |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4454 <lb/><lb/></s></p></div> |
|
22d6a63640c6
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|
4455 <p xml:lang="lat"> |
|
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|
4456 <s xml:id="echoid-s372" 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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|
4457 Inquisitio indentata hasta apud Stratford Langthornes |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4458 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4459 </p> |
|
22d6a63640c6
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|
4460 <p xml:lang="lat"> |
|
22d6a63640c6
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|
4461 <s xml:id="echoid-s373" xml:space="preserve"> |
|
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|
4462 Bis non spondebis <lb/> |
|
22d6a63640c6
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|
4463 quod præsto solvere valebis |
|
22d6a63640c6
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|
4464 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4465 </p> |
|
22d6a63640c6
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|
4466 <p xml:lang="lat"> |
|
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|
4467 <s xml:id="echoid-s374" 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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|
4468 In Principio |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4469 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4470 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4471 <p xml:lang="lat"> |
|
22d6a63640c6
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|
4472 <s xml:id="echoid-s375" 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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|
4473 Bis duo notam |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4474 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4475 </p> |
|
22d6a63640c6
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|
4476 <p xml:lang="lat"> |
|
22d6a63640c6
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|
4477 <s xml:id="echoid-s376" xml:space="preserve"> |
|
22d6a63640c6
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|
4478 Arbor ut ex fractu <lb/> |
|
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|
4479 sit nequam noscitur acta |
|
22d6a63640c6
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|
4480 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4481 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4482 <p> |
|
22d6a63640c6
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|
4483 <s xml:id="echoid-s377" 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
|
4484 James |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4485 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4486 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4487 <p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4488 <s xml:id="echoid-s378" 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
|
4489 Alexand |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
4490 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4491 </p> |
|
22d6a63640c6
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|
4492 <p> |
|
22d6a63640c6
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|
4493 <s xml:id="echoid-s379" 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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|
4494 cadaeibf <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
|
4495 31415926 |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
4496 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4497 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4498 <pb file="add_6787_f319" o="319" n="636"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
4499 <p><s xml:id="echoid-s380"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi><mi>d</mi><mi>e</mi></mstyle></math></s></p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4500 <p><s xml:id="echoid-s381"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>1</mn></mstyle></math><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
|
4501 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>,</mo><mi>a</mi><mi>c</mi><mo>,</mo><mi>z</mi><mi>x</mi></mstyle></math></s><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
|
4502 <s xml:id="echoid-s382"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mo>,</mo><mi>z</mi><mi>x</mi><mo>:</mo><mi>a</mi><mi>b</mi><mi>e</mi><mo>,</mo><mi>a</mi><mi>c</mi><mi>d</mi></mstyle></math><lb/> |
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4503 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>2</mn></mstyle></math></s></p> |
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4504 <p><s xml:id="echoid-s383"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mo>,</mo><mi>a</mi><mi>b</mi><mo>,</mo><mi>z</mi><mi>x</mi></mstyle></math></s><lb/> |
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4505 <s xml:id="echoid-s384"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mo>,</mo><mi>z</mi><mi>x</mi><mo>:</mo><mi>a</mi><mi>c</mi><mi>d</mi><mo>,</mo><mi>a</mi><mi>b</mi><mi>e</mi></mstyle></math><lb/> |
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4506 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>1</mn></mstyle></math></s></p> |
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4507 <p><s xml:id="echoid-s385"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mo>-</mo><mo>√</mo><mn>2</mn><mo>,</mo><mo>√</mo><mn>2</mn><mo>,</mo><mn>1</mn></mstyle></math></s></p> |
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4508 <p><s xml:id="echoid-s386"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo>-</mo><mn>1</mn><mn>4</mn><mn>1</mn><mn>4</mn><mo>=</mo><mn>5</mn><mn>8</mn><mn>6</mn></mstyle></math></s></p> |
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4509 <p><s xml:id="echoid-s387"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>4</mn><mn>1</mn><mn>4</mn><mo maxsize="1.01">/</mo><mn>5</mn><mn>8</mn><mn>6</mn><mo>=</mo><mn>2</mn><mfrac><mrow><mn>2</mn></mrow><mrow><mn>5</mn></mrow></mfrac></mstyle></math></s></p> |
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4510 <pb file="add_6787_f319v" o="319v" n="637"/> |
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4511 <p><s xml:id="echoid-s388"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>3</mn></mstyle></math></s></p> |
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4512 <p><s xml:id="echoid-s389"><reg norm="Tempus" type="abbr">Temp.</reg> <reg norm="Spatium" type="abbr">Spac.</reg></s></p> |
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4513 <p><s xml:id="echoid-s390"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mn>0</mn><mn>0</mn><mo>×</mo><mn>5</mn><mo>=</mo><mn>1</mn><mn>5</mn><mn>0</mn><mn>0</mn></mstyle></math></s></p> |
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4514 <pb file="add_6787_f320" o="320" n="638"/> |
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4515 <pb file="add_6787_f320v" o="320v" n="639"/> |
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4516 <pb file="add_6787_f321" o="321" n="640"/> |
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4517 <pb file="add_6787_f321v" o="321v" n="641"/> |
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4518 <pb file="add_6787_f322" o="322" n="642"/> |
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4519 <pb file="add_6787_f322v" o="322v" n="643"/> |
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4520 <pb file="add_6787_f323" o="323" n="644"/> |
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4521 <pb file="add_6787_f323v" o="323v" n="645"/> |
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4522 <pb file="add_6787_f324" o="324" n="646"/> |
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4523 <p><s xml:id="echoid-s391"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>1</mn><mn>0</mn><mn>1</mn><mn>0</mn><mn>8</mn><mn>8</mn><mn>9</mn><mn>2</mn><mn>8</mn><mn>6</mn><mn>0</mn><mn>5</mn><mn>1</mn><mn>7</mn><mn>0</mn><mn>0</mn><mn>4</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>5</mn><mn>4</mn><mn>2</mn><mn>9</mn><mn>9</mn><mn>0</mn><mn>1</mn><mn>1</mn><mn>1</mn><mn>2</mn><mo>.</mo><mn>8</mn><mn>0</mn><mn>2</mn><mn>7</mn></mstyle></math></s></p> |
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4524 <p><s xml:id="echoid-s392"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>5</mn><mn>4</mn><mn>2</mn><mn>9</mn><mn>9</mn><mn>0</mn><mn>1</mn><mn>1</mn><mn>1</mn><mn>2</mn><mn>8</mn><mn>0</mn><mn>2</mn><mn>7</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>2</mn><mn>7</mn><mn>1</mn><mn>1</mn><mn>2</mn><mn>7</mn><mn>5</mn><mn>0</mn><mn>5</mn><mn>0</mn><mo>.</mo><mn>2</mn><mn>0</mn><mn>2</mn><mn>4</mn></mstyle></math></s></p> |
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4525 <p><s xml:id="echoid-s393"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>2</mn><mn>7</mn><mn>1</mn><mn>1</mn><mn>2</mn><mn>7</mn><mn>5</mn><mn>0</mn><mn>5</mn><mn>0</mn><mn>2</mn><mn>0</mn><mn>2</mn><mn>4</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>1</mn><mn>3</mn><mn>5</mn><mn>4</mn><mn>7</mn><mn>1</mn><mn>9</mn><mn>8</mn><mn>9</mn><mn>2</mn><mo>.</mo><mn>1</mn><mn>0</mn><mn>8</mn><mn>1</mn></mstyle></math></s></p> |
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4526 <p><s xml:id="echoid-s394"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>1</mn><mn>3</mn><mn>5</mn><mn>4</mn><mn>7</mn><mn>1</mn><mn>9</mn><mn>8</mn><mn>9</mn><mn>2</mn><mn>1</mn><mn>0</mn><mn>8</mn><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>6</mn><mn>7</mn><mn>7</mn><mn>1</mn><mn>3</mn><mn>0</mn><mn>6</mn><mn>9</mn><mn>3</mn><mo>.</mo><mn>0</mn><mn>6</mn><mn>6</mn><mn>3</mn></mstyle></math></s></p> |
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4527 <p><s xml:id="echoid-s395"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>6</mn><mn>7</mn><mn>7</mn><mn>1</mn><mn>3</mn><mn>0</mn><mn>6</mn><mn>9</mn><mn>3</mn><mn>0</mn><mn>6</mn><mn>6</mn><mn>3</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>3</mn><mn>3</mn><mn>8</mn><mn>5</mn><mn>0</mn><mn>8</mn><mn>0</mn><mn>5</mn><mn>2</mn><mo>.</mo><mn>6</mn><mn>8</mn><mn>2</mn><mn>2</mn></mstyle></math></s></p> |
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4528 <p><s xml:id="echoid-s396"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>3</mn><mn>3</mn><mn>8</mn><mn>5</mn><mn>0</mn><mn>8</mn><mn>0</mn><mn>5</mn><mn>2</mn><mn>6</mn><mn>8</mn><mn>2</mn><mn>2</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>1</mn><mn>6</mn><mn>9</mn><mn>2</mn><mn>3</mn><mn>9</mn><mn>7</mn><mn>0</mn><mn>5</mn><mo>.</mo><mn>3</mn><mn>0</mn><mn>2</mn><mn>1</mn></mstyle></math></s></p> |
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4529 <pb file="add_6787_f324v" o="324v" n="647"/> |
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4530 <pb file="add_6787_f325" o="325" n="648"/> |
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4531 <p><s xml:id="echoid-s397"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mn>8</mn><mn>5</mn><mn>7</mn><mn>8</mn><mn>7</mn><mn>9</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo maxsize="1.01">/</mo><mn>1</mn><mn>4</mn><mn>1</mn><mn>4</mn><mn>2</mn><mn>1</mn><mn>3</mn><mo>=</mo><mn>4</mn><mn>1</mn><mn>4</mn><mn>2</mn><mn>1</mn></mstyle></math></s></p> |
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4532 <p><s xml:id="echoid-s398"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>1</mn><mn>4</mn><mn>1</mn><mn>4</mn><mn>2</mn><mn>1</mn><mn>3</mn><mn>5</mn><mn>6</mn><mn>2</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>1</mn><mn>8</mn><mn>9</mn><mn>2</mn><mn>0</mn><mn>7</mn><mo>.</mo><mn>1</mn></mstyle></math></s></p> |
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4533 <p><s xml:id="echoid-s399"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>D</mi><mi>c</mi></mstyle></math><lb/> |
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4534 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>2</mn><mo>-</mo><mn>1</mn><mi>e</mi><mi>t</mi></mstyle></math><lb/> |
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4535 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>×</mo><mn>2</mn></mstyle></math><lb/> |
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4536 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>8</mn><mo>-</mo><mn>2</mn></mstyle></math></s><lb/> |
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4537 <s xml:id="echoid-s400"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><mn>4</mn><mo>√</mo></mrow><mn>2</mn><mo>-</mo><mn>1</mn><mi>g</mi><mi>y</mi></mstyle></math><lb/> |
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4538 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>×</mo><mn>4</mn></mstyle></math><lb/> |
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4539 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><mn>4</mn><mo>√</mo></mrow><mn>5</mn><mn>1</mn><mn>2</mn><mo>-</mo><mn>4</mn></mstyle></math></s><lb/> |
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4540 <s xml:id="echoid-s401"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><mn>8</mn><mo>√</mo></mrow><mn>2</mn><mo>-</mo><mn>1</mn><mi>r</mi><mi>H</mi></mstyle></math><lb/> |
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4541 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>×</mo><mn>8</mn></mstyle></math><lb/> |
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4542 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><mn>8</mn><mo>√</mo></mrow><mn>3</mn><mn>3</mn><mn>5</mn><mn>5</mn><mn>4</mn><mn>4</mn><mn>3</mn><mn>2</mn><mo>-</mo><mn>8</mn></mstyle></math></s><lb/></p> |
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4543 <p><s xml:id="echoid-s402"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mn>1</mn><mn>4</mn><mo>×</mo><mn>4</mn><mn>1</mn><mn>4</mn><mo>=</mo><mn>1</mn><mn>7</mn><mn>1</mn><mn>3</mn><mn>9</mn><mn>6</mn></mstyle></math></s></p> |
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4544 <p><s xml:id="echoid-s403"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mn>4</mn><mo>×</mo><mn>8</mn><mo>=</mo><mn>5</mn><mn>1</mn><mn>2</mn></mstyle></math></s></p> |
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4545 <p><s xml:id="echoid-s404"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mo>×</mo><mn>4</mn><mo>=</mo><mn>1</mn><mn>6</mn></mstyle></math></s></p> |
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4546 <p><s xml:id="echoid-s405"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>6</mn><mo>×</mo><mn>1</mn><mn>6</mn><mo>=</mo><mn>2</mn><mn>5</mn><mn>6</mn></mstyle></math></s></p> |
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4547 <p><s xml:id="echoid-s406"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>5</mn><mn>6</mn><mo>×</mo><mn>2</mn><mo>=</mo><mn>5</mn><mn>1</mn><mn>2</mn></mstyle></math></s></p> |
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4548 <p><s xml:id="echoid-s407"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>8</mn><mo>×</mo><mn>8</mn><mo>=</mo><mn>6</mn><mn>4</mn></mstyle></math></s></p> |
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4549 <p><s xml:id="echoid-s408"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mn>4</mn><mo>×</mo><mn>6</mn><mn>4</mn><mo>=</mo><mn>4</mn><mn>0</mn><mn>9</mn><mn>6</mn></mstyle></math></s></p> |
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4550 <p><s xml:id="echoid-s409"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn><mn>0</mn><mn>9</mn><mn>6</mn><mo>×</mo><mn>4</mn><mn>0</mn><mn>9</mn><mn>6</mn><mo>=</mo><mn>1</mn><mn>6</mn><mn>7</mn><mn>7</mn><mn>7</mn><mn>2</mn><mn>1</mn><mn>6</mn></mstyle></math></s></p> |
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4551 <p><s xml:id="echoid-s410"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>6</mn><mn>7</mn><mn>7</mn><mn>7</mn><mn>2</mn><mn>1</mn><mn>6</mn><mo>+</mo><mn>1</mn><mn>6</mn><mn>7</mn><mn>7</mn><mn>7</mn><mn>2</mn><mn>1</mn><mn>6</mn><mo>=</mo><mn>3</mn><mn>3</mn><mn>5</mn><mn>5</mn><mn>4</mn><mn>4</mn><mn>3</mn><mn>2</mn></mstyle></math></s></p> |
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4552 <p><s xml:id="echoid-s411"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mn>1</mn><mn>2</mn><mo>×</mo><mn>5</mn><mn>1</mn><mn>2</mn><mo>=</mo><mn>2</mn><mn>6</mn><mn>2</mn><mn>1</mn><mn>4</mn><mn>4</mn></mstyle></math></s></p> |
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4553 <p><s xml:id="echoid-s412"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>6</mn><mn>2</mn><mn>1</mn><mn>4</mn><mn>4</mn><mo>×</mo><mn>5</mn><mn>1</mn><mn>2</mn><mo>=</mo><mn>1</mn><mn>3</mn><mn>4</mn><mn>2</mn><mn>1</mn><mn>7</mn><mn>7</mn><mn>2</mn><mn>8</mn></mstyle></math></s></p> |
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4554 <p><s xml:id="echoid-s413"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>1</mn><mn>1</mn><mn>8</mn><mn>9</mn><mn>2</mn><mn>0</mn><mn>7</mn><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>1</mn><mn>0</mn><mn>9</mn><mn>0</mn><mn>5</mn><mn>0</mn><mn>7</mn><mo>.</mo><mn>7</mn></mstyle></math></s></p> |
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4555 <p><s xml:id="echoid-s414"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>9</mn><mn>0</mn><mn>5</mn><mn>0</mn><mn>7</mn><mn>7</mn><mo>×</mo><mn>8</mn><mo>=</mo><mn>7</mn><mn>2</mn><mn>4</mn><mn>0</mn><mn>6</mn><mn>1</mn><mn>6</mn></mstyle></math></s></p> |
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4556 <p><s xml:id="echoid-s415"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>8</mn><mo>-</mo><mn>2</mn></mstyle></math></s></p> |
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4557 <p><s xml:id="echoid-s416"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mo>√</mo><mn>2</mn><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo><mo>×</mo><mn>2</mn></mstyle></math></s></p> |
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4558 <p><s xml:id="echoid-s417"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mrow><mn>4</mn><mo>√</mo></mrow><mn>2</mn><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo><mo>×</mo><mn>4</mn></mstyle></math></s></p> |
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4559 <p><s xml:id="echoid-s418"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mrow><mn>8</mn><mo>√</mo></mrow><mn>2</mn><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo><mo>×</mo><mn>8</mn></mstyle></math></s></p> |
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4560 <p><s xml:id="echoid-s419"><emph style="st"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>2</mn><mo>-</mo><mn>1</mn></mstyle></math><lb/> |
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4561 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>8</mn><mo>-</mo><mn>2</mn></mstyle></math></emph></s></p> |
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4562 <p><s xml:id="echoid-s420"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>√</mo><mn>8</mn><mo>-</mo><mn>2</mn></mstyle></math></s></p> |
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4563 <p><s xml:id="echoid-s421"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><mn>4</mn><mo>√</mo></mrow><mn>5</mn><mn>1</mn><mn>2</mn><mo>-</mo><mn>4</mn></mstyle></math></s></p> |
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4564 <p><s xml:id="echoid-s422"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><mn>8</mn><mo>√</mo></mrow><mn>3</mn><mn>3</mn><mn>5</mn><mn>5</mn><mn>4</mn><mn>4</mn><mn>3</mn><mn>2</mn><mo>-</mo><mn>8</mn></mstyle></math></s></p> |
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4565 <p><s xml:id="echoid-s423">Quæritur Summa</s></p> |
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4566 <pb file="add_6787_f325v" o="325v" n="649"/> |
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4567 <pb file="add_6787_f326" o="326" n="650"/> |
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4568 <p><s xml:id="echoid-s424"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mn>5</mn><mo>,</mo><mn>2</mn><mn>2</mn><mo>,</mo><mn>1</mn><mn>6</mn></mstyle></math></s></p> |
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4569 <pb file="add_6787_f326v" o="326v" n="651"/> |
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4570 <pb file="add_6787_f327" o="327" n="652"/> |
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4571 <pb file="add_6787_f327v" o="327v" n="653"/> |
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4572 <pb file="add_6787_f328" o="328" n="654"/> |
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4573 <pb file="add_6787_f328v" o="328v" n="655"/> |
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4574 <pb file="add_6787_f329" o="329" n="656"/> |
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4575 <pb file="add_6787_f329v" o="329v" n="657"/> |
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4576 <pb file="add_6787_f330" o="330" n="658"/> |
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4577 <pb file="add_6787_f330v" o="330v" n="659"/> |
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4578 <pb file="add_6787_f331" o="331" n="660"/> |
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4579 <pb file="add_6787_f331v" o="331v" n="661"/> |
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4580 <pb file="add_6787_f332" o="332" n="662"/> |
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4581 <pb file="add_6787_f332v" o="332v" n="663"/> |
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4582 <pb file="add_6787_f333" o="333" n="664"/> |
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4583 <pb file="add_6787_f333v" o="333v" n="665"/> |
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4584 <pb file="add_6787_f334" o="334" n="666"/> |
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4585 <pb file="add_6787_f334v" o="334v" n="667"/> |
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4586 <pb file="add_6787_f335" o="335" n="668"/> |
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4587 <pb file="add_6787_f335v" o="335v" n="669"/> |
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4588 <pb file="add_6787_f336" o="336" n="670"/> |
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4589 <pb file="add_6787_f336v" o="336v" n="671"/> |
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4590 <pb file="add_6787_f337" o="337" n="672"/> |
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4591 <pb file="add_6787_f337v" o="337v" n="673"/> |
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4592 <pb file="add_6787_f338" o="338" n="674"/> |
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4593 <pb file="add_6787_f338v" o="338v" n="675"/> |
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4594 <div xml:id="echoid-div87" type="page_commentary" level="2" n="87"> |
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4595 <p> |
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4596 <s xml:id="echoid-s425" xml:space="preserve">[<emph style="it">Note: |
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4597 <p> |
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4598 <s xml:id="echoid-s425" xml:space="preserve"> |
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4599 Calulations based on on the series 3, 5, 7, 9, ..., with constant difference 2. <lb/> |
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4600 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> denote sums of terms taken two, three, four, or five at a time. |
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4601 </s> |
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4602 </p> |
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4603 </emph>] |
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4604 <lb/><lb/></s></p></div> |
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4605 <head xml:id="echoid-head87" xml:space="preserve" xml:lang="lat"> |
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4606 4. Differentiæ differentiarum |
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4607 <lb/>[<emph style="it">tr: |
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|
4608 Differences of differences |
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4609 </emph>]<lb/> |
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4610 </head> |
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4611 <pb file="add_6787_f339" o="339" n="676"/> |
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4612 <div xml:id="echoid-div88" type="page_commentary" level="2" n="88"> |
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4613 <p> |
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4614 <s xml:id="echoid-s427" xml:space="preserve">[<emph style="it">Note: |
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4615 <p> |
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4616 <s xml:id="echoid-s427" xml:space="preserve"> |
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4617 Calculations based on a general series with constant second difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/> |
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4618 The second and first rows begin with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> respectively. <lb/> |
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4619 The series 5, 8, 13, 20, ... is used as an example, thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mo>=</mo><mn>5</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mn>3</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>. |
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4620 (A second version takes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mo>=</mo><mn>5</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mn>1</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>.) |
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4621 <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> denote successive sums of terms, taken two at a time. |
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4622 </s> |
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4623 </p> |
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4624 </emph>] |
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4625 <lb/><lb/></s></p></div> |
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4626 <head xml:id="echoid-head88" xml:space="preserve" xml:lang="lat"> |
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4627 5. Differentiæ differentiarum differentiarum |
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4628 <lb/>[<emph style="it">tr: |
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4629 Differences of differences of differences |
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4630 </emph>]<lb/> |
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4631 </head> |
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4632 <p xml:lang="lat"> |
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4633 <s xml:id="echoid-s429" xml:space="preserve"> |
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4634 differentiæ |
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4635 <lb/>[<emph style="it">tr: |
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4636 differences |
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4637 </emph>]<lb/> |
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4638 </s> |
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4639 </p> |
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4640 <p xml:lang="lat"> |
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4641 <s xml:id="echoid-s430" xml:space="preserve"> |
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4642 Aliter |
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4643 <lb/>[<emph style="it">tr: |
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4644 Another way |
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4645 </emph>]<lb/> |
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4646 </s> |
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4647 </p> |
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4648 <pb file="add_6787_f339v" o="339v" n="677"/> |
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4649 <pb file="add_6787_f340" o="340" n="678"/> |
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4650 <pb file="add_6787_f340v" o="340v" n="679"/> |
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4651 <div xml:id="echoid-div89" type="page_commentary" level="2" n="89"> |
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4652 <p> |
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4653 <s xml:id="echoid-s431" xml:space="preserve">[<emph style="it">Note: |
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4654 <p> |
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4655 <s xml:id="echoid-s431" xml:space="preserve"> |
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4656 Further calculations on the series 5, 8, 13, 20, ... from Add MS 6787, f. 339. <lb/> |
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4657 <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> denote successive sums of terms, taken two at a time. <lb/> |
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4658 Fingers in the margin point to formulae for <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>e</mi></mstyle></math> in terms of <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>. |
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4659 </s> |
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4660 </p> |
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4661 </emph>] |
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4662 <lb/><lb/></s></p></div> |
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4663 <head xml:id="echoid-head89" xml:space="preserve"> |
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4664 5).2. |
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4665 </head> |
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4666 <pb file="add_6787_f341" o="341" n="680"/> |
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4667 <div xml:id="echoid-div90" type="page_commentary" level="2" n="90"> |
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4668 <p> |
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4669 <s xml:id="echoid-s433" xml:space="preserve">[<emph style="it">Note: |
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4670 <p> |
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4671 <s xml:id="echoid-s433" xml:space="preserve"> |
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4672 Further calculations on the series 5, 8, 13, 20, ... from Add MS 6787, f. 339. <lb/> |
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4673 <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> denote successive sums of terms, taken two at a time. <lb/> |
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4674 A finger in the margin points to a formula for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> in terms of <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>. |
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4675 </s> |
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4676 </p> |
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4677 </emph>] |
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4678 <lb/><lb/></s></p></div> |
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4679 <head xml:id="echoid-head90" xml:space="preserve"> |
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4680 5).3. |
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4681 </head> |
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4682 <pb file="add_6787_f341v" o="341v" n="681"/> |
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4683 <div xml:id="echoid-div91" type="page_commentary" level="2" n="91"> |
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4684 <p> |
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4685 <s xml:id="echoid-s435" xml:space="preserve">[<emph style="it">Note: |
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4686 <p> |
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4687 <s xml:id="echoid-s435" xml:space="preserve"> |
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4688 Further calculations on the series 5, 8, 13, 20, ... from Add MS 6787, f. 339. <lb/> |
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4689 <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> denote successive sums of terms, taken two at a time. <lb/> |
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4690 Fingers in the margin point to formulae for <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>e</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> in terms of <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>. |
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4691 </s> |
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4692 </p> |
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4693 </emph>] |
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4694 <lb/><lb/></s></p></div> |
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4695 <head xml:id="echoid-head91" xml:space="preserve"> |
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4696 5).4. |
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4697 </head> |
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4698 <pb file="add_6787_f342" o="342" n="682"/> |
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4699 <div xml:id="echoid-div92" type="page_commentary" level="2" n="92"> |
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4700 <p> |
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4701 <s xml:id="echoid-s437" xml:space="preserve">[<emph style="it">Note: |
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4702 <p> |
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4703 <s xml:id="echoid-s437" xml:space="preserve"> |
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4704 Calculations similar to those in Add MS 6787, f. 339. <lb/> |
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4705 Here the series 7, 9, 14, 32, ... is used as an example, thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mo>=</mo><mn>7</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mn>2</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn></mstyle></math>. <lb/> |
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4706 The second example is 5, 8, 13, ..., as on Add MS 6787, f. 339. <lb/> |
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4707 <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> denote successive sums of terms, taken three at a time. |
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4708 </s> |
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4709 </p> |
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4710 </emph>] |
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4711 <lb/><lb/></s></p></div> |
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4712 <head xml:id="echoid-head92" xml:space="preserve" xml:lang="lat"> |
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4713 6. Differentiæ differentiarum differentiarum |
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4714 <lb/>[<emph style="it">tr: |
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4715 Differences of differences of differences |
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4716 </emph>]<lb/> |
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4717 </head> |
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4718 <pb file="add_6787_f342v" o="342v" n="683"/> |
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4719 <pb file="add_6787_f343" o="343" n="684"/> |
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4720 <pb file="add_6787_f343v" o="343v" n="685"/> |
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4721 <div xml:id="echoid-div93" type="page_commentary" level="2" n="93"> |
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4722 <p> |
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4723 <s xml:id="echoid-s439" xml:space="preserve">[<emph style="it">Note: |
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4724 <p> |
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4725 <s xml:id="echoid-s439" xml:space="preserve"> |
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4726 Calculations similar to those on Add MS 6787, f. 341v, now for the series 7, 9, 14, ...from Add MS 6787, f. 342. <lb/> |
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4727 <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> denote successive sums of terms, taken three at a time. <lb/> |
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4728 Fingers in the margin point to formulae for <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>e</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> in terms of <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>. |
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4729 </s> |
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4730 </p> |
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4731 </emph>] |
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4732 <lb/><lb/></s></p></div> |
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4733 <head xml:id="echoid-head93" xml:space="preserve"> |
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4734 6). |
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4735 </head> |
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4736 <pb file="add_6787_f344" o="344" n="686"/> |
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4737 <pb file="add_6787_f344v" o="344v" n="687"/> |
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4738 <div xml:id="echoid-div94" type="page_commentary" level="2" n="94"> |
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4739 <p> |
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4740 <s xml:id="echoid-s441" xml:space="preserve">[<emph style="it">Note: |
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4741 <p> |
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4742 <s xml:id="echoid-s441" xml:space="preserve"> |
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4743 Further calculations for the series 5, 8, 13, ... from Add MS 6787, f. 339. <lb/> |
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4744 <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> denote successive sums of terms, taken four at a time. <lb/> |
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4745 Fingers in the margin point to formulae for <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>e</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> in terms of <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>. |
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4746 </s> |
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4747 </p> |
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4748 </emph>] |
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4749 <lb/><lb/></s></p></div> |
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4750 <head xml:id="echoid-head94" xml:space="preserve" xml:lang="lat"> |
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4751 7.) Differentiæ differentiarum differentiarum |
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4752 <lb/>[<emph style="it">tr: |
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4753 Differences of differences of differences |
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4754 </emph>]<lb/> |
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4755 </head> |
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4756 <pb file="add_6787_f345" o="345" n="688"/> |
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4757 <div xml:id="echoid-div95" type="page_commentary" level="2" n="95"> |
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4758 <p> |
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4759 <s xml:id="echoid-s443" xml:space="preserve">[<emph style="it">Note: |
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4760 <p> |
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4761 <s xml:id="echoid-s443" xml:space="preserve"> |
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4762 Further calculations for the series 5, 8, 13, ... from Add MS 6787, f. 339. <lb/> |
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4763 <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> denote successive sums of terms, taken five at a time. <lb/> |
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4764 Fingers in the margin point to formulae for <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>e</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> in terms of <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>. |
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4765 </s> |
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4766 </p> |
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4767 </emph>] |
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4768 <lb/><lb/></s></p></div> |
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4769 <head xml:id="echoid-head95" xml:space="preserve" xml:lang="lat"> |
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4770 8.) Differentiæ differentiarum differentiarum |
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4771 <lb/>[<emph style="it">tr: |
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4772 Differences of differences of differences |
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4773 </emph>]<lb/> |
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4774 </head> |
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4775 <pb file="add_6787_f345v" o="345v" n="689"/> |
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4776 <pb file="add_6787_f346" o="346" n="690"/> |
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4777 <div xml:id="echoid-div96" type="page_commentary" level="2" n="96"> |
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4778 <p> |
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4779 <s xml:id="echoid-s445" xml:space="preserve">[<emph style="it">Note: |
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4780 <p> |
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4781 <s xml:id="echoid-s445" xml:space="preserve"> |
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4782 Calculations on a general series with constant third difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/> |
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4783 The third, second, and first rows begin with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math> respectively. <lb/> |
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4784 The series 10, 15, 23, ... is used as an example, thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi><mo>=</mo><mn>1</mn><mn>0</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mo>=</mo><mn>5</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mo>=</mo><mn>3</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn></mstyle></math>. <lb/> |
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4785 <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> denote successive sums of terms, taken two at a time. <lb/> |
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4786 </s> |
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4787 </p> |
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4788 </emph>] |
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4789 <lb/><lb/></s></p></div> |
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4790 <head xml:id="echoid-head96" xml:space="preserve" xml:lang="lat"> |
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4791 9) Differentiæ differentiarum differentiarum differentiarum |
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4792 <lb/>[<emph style="it">tr: |
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4793 Differences of differences of differences of differences |
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4794 </emph>]<lb/> |
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4795 </head> |
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4796 <pb file="add_6787_f346v" o="346v" n="691"/> |
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4797 <pb file="add_6787_f347" o="347" n="692"/> |
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4798 <pb file="add_6787_f347v" o="347v" n="693"/> |
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4799 <div xml:id="echoid-div97" type="page_commentary" level="2" n="97"> |
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4800 <p> |
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4801 <s xml:id="echoid-s447" xml:space="preserve">[<emph style="it">Note: |
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4802 <p> |
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4803 <s xml:id="echoid-s447" xml:space="preserve"> |
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4804 Further calculations for the series 10, 15, 23, ... from Add MS 6787, f. 346. <lb/> |
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4805 <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> denote successive sums of terms, taken two at a time. <lb/> |
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4806 The page shows formulae for <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>e</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> in terms of <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>. |
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4807 </s> |
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4808 </p> |
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4809 </emph>] |
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4810 <lb/><lb/></s></p></div> |
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4811 <head xml:id="echoid-head97" xml:space="preserve"> |
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4812 9) |
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4813 </head> |
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4814 <pb file="add_6787_f348" o="348" n="694"/> |
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4815 <pb file="add_6787_f348v" o="348v" n="695"/> |
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4816 <div xml:id="echoid-div98" type="page_commentary" level="2" n="98"> |
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4817 <p> |
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4818 <s xml:id="echoid-s449" xml:space="preserve">[<emph style="it">Note: |
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4819 <p> |
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4820 <s xml:id="echoid-s449" xml:space="preserve"> |
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4821 Further calculations for the series 10, 15, 23, ... from Add MS 6787, f. 346. <lb/> |
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4822 <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> denote successive sums of terms, taken three at a time. <lb/> |
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4823 The page shows formulae for <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>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. |
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4824 </s> |
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4825 </p> |
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4826 </emph>] |
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4827 <lb/><lb/></s></p></div> |
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4828 <head xml:id="echoid-head98" xml:space="preserve" xml:lang="lat"> |
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4829 10) Differentiæ differentiarum differentiarum differentiarum |
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4830 <lb/>[<emph style="it">tr: |
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4831 Differences of differences of differences of differences |
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4832 </emph>]<lb/> |
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4833 </head> |
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4834 <pb file="add_6787_f349" o="349" n="696"/> |
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4835 <pb file="add_6787_f349v" o="349v" n="697"/> |
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4836 <div xml:id="echoid-div99" type="page_commentary" level="2" n="99"> |
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4837 <p> |
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4838 <s xml:id="echoid-s451" xml:space="preserve">[<emph style="it">Note: |
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4839 <p> |
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4840 <s xml:id="echoid-s451" xml:space="preserve"> |
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4841 Further calculations for the series 10, 15, 23, ... from Add MS 6787, f. 346. <lb/> |
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4842 <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> denote successive sums of terms, taken four at a time. <lb/> |
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4843 The page shows formulae for <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>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. |
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4844 </s> |
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4845 </p> |
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4846 </emph>] |
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4847 <lb/><lb/></s></p></div> |
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4848 <head xml:id="echoid-head99" xml:space="preserve" xml:lang="lat"> |
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4849 11) Differentiæ differentiarum differentiarum differentiarum |
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4850 <lb/>[<emph style="it">tr: |
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4851 Differences of differences of differences of differences |
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4852 </emph>]<lb/> |
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4853 </head> |
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4854 <pb file="add_6787_f350" o="350" n="698"/> |
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4855 <pb file="add_6787_f350v" o="350v" n="699"/> |
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4856 <pb file="add_6787_f351" o="351" n="700"/> |
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4857 <pb file="add_6787_f351v" o="351v" n="701"/> |
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4858 <pb file="add_6787_f352" o="352" n="702"/> |
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4859 <pb file="add_6787_f352v" o="352v" n="703"/> |
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4860 <pb file="add_6787_f353" o="353" n="704"/> |
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4861 <pb file="add_6787_f353v" o="353v" n="705"/> |
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4862 <pb file="add_6787_f354" o="354" n="706"/> |
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4863 <pb file="add_6787_f354v" o="354v" n="707"/> |
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4864 <pb file="add_6787_f355" o="355" n="708"/> |
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4865 <pb file="add_6787_f355v" o="355v" n="709"/> |
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4866 <pb file="add_6787_f356" o="356" n="710"/> |
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4867 <pb file="add_6787_f356v" o="356v" n="711"/> |
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4868 <pb file="add_6787_f357" o="357" n="712"/> |
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4869 <div xml:id="echoid-div100" type="page_commentary" level="2" n="100"> |
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4870 <p> |
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4871 <s xml:id="echoid-s453" xml:space="preserve">[<emph style="it">Note: |
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4872 <p> |
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4873 <s xml:id="echoid-s453" xml:space="preserve"> |
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4874 This is the first of a set of 11 pages containing propositions from Book I of the |
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4875 <emph style="it">Conics</emph> of Apollonius. The edition used by Harriot was Commandino's |
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4876 <emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566), |
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4877 but Harriot translated the verbal propositions and proofs into his own symbolic notation. <lb/> |
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4878 English translations of the propositions are taken from Dana Densmore (ed), |
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4879 <emph style="it">Apollonius of Perga: Conics Books I–III</emph>, Green Lion Press, 1998. <lb/> |
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4880 </s> |
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4881 <lb/> |
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4882 <s xml:id="echoid-s454" xml:space="preserve"> |
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4883 Proposition 11 of Book I is Apollonius's definition of a parabola. |
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4884 </s> |
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4885 <lb/> |
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4886 <quote> |
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4887 I.11 |
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4888 If a cone is cut by a plane through its axis, and also cut by another plane |
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4889 cutting the base of the cone in a straight line perpendicular to the base of the axial triangle, |
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4890 and if, further, the diameter of the section is parallel to one side of the axial triangle, |
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4891 and if any straight line is drawn from the section of the cone to its diameter |
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4892 such that this straight line is parallel to the common section of the cutting plane and of the cone 19s base, |
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4893 then this straight line to the diameter will equal in square the rectangle contained |
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4894 by the straight line from the section 19s vertex to where the straight line to the diameter cuts it off, |
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4895 and another straight line which has the same ratio to the straight line between the angle of the cone |
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4896 and the vertex of the section as the square on the base of the axial triangle |
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4897 has to the rectangle contained by the remaining two sides of the triangle. |
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4898 And let such a section be called a parabola. |
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4899 </quote> |
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4900 </p> |
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4901 </emph>] |
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4902 <lb/><lb/></s></p></div> |
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4903 <head xml:id="echoid-head100" xml:space="preserve" xml:lang="lat"> |
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4904 Appol. lib. 1. prop. 11. De conicis |
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4905 <lb/>[<emph style="it">tr: |
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4906 Apollonius, Book I, Proposition 11. On conics |
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4907 </emph>]<lb/> |
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4908 </head> |
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4909 <p xml:lang="lat"> |
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4910 <s xml:id="echoid-s456" xml:space="preserve"> |
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4911 parabola |
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4912 </s> |
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4913 </p> |
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4914 <p xml:lang="lat"> |
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4915 <s xml:id="echoid-s457" xml:space="preserve"> |
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4916 Aliter demonstrationem <lb/> |
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4917 ordinavimus ut <lb/> |
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4918 sequitur. |
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4919 <lb/>[<emph style="it">tr: |
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4920 Another demonstration, which we have ordered as follows. |
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4921 </emph>]<lb/> |
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4922 </s> |
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4923 </p> |
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4924 <p xml:lang="lat"> |
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4925 <s xml:id="echoid-s458" xml:space="preserve"> |
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4926 Ratio componitur ex |
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4927 <lb/>[<emph style="it">tr: |
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4928 Ratio composed from |
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4929 </emph>]<lb/> |
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4930 </s> |
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4931 </p> |
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4932 <p xml:lang="lat"> |
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4933 <s xml:id="echoid-s459" xml:space="preserve"> |
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4934 Quæsitum |
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4935 <lb/>[<emph style="it">tr: |
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4936 Sought |
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4937 </emph>]<lb/> |
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4938 </s> |
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4939 </p> |
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4940 <pb file="add_6787_f357v" o="357v" n="713"/> |
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4941 <pb file="add_6787_f358" o="358" n="714"/> |
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4942 <div xml:id="echoid-div101" type="page_commentary" level="2" n="101"> |
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4943 <p> |
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4944 <s xml:id="echoid-s460" xml:space="preserve">[<emph style="it">Note: |
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4945 <p> |
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4946 <s xml:id="echoid-s460" xml:space="preserve"> |
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|
4947 Proposition 20 of Book I of Apollonius, edited by Commandino in |
|
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4948 <emph style="it">Conicorum libri quattuor</emph> (1566), |
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4949 is the defining property of a parabola in terms of its ordinates to the diameter. |
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|
4950 </s> |
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4951 <lb/> |
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4952 <quote> |
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4953 I.20 |
|
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4954 If in a parabola two straight lines are dropped as ordinates to the diameter, |
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4955 the squares on them will be to each other as the straight lines cut off by them on the diameter |
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4956 beginning from the vertex are to each other. |
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|
4957 </quote> |
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4958 </p> |
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4959 </emph>] |
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4960 <lb/><lb/></s></p></div> |
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|
4961 <head xml:id="echoid-head101" xml:space="preserve" xml:lang="lat"> |
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|
4962 Appol. lib. 1. prop. 20. De conicis |
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4963 <lb/>[<emph style="it">tr: |
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|
4964 Apollonius, Book I, Proposition 20. On conics |
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4965 </emph>]<lb/> |
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4966 </head> |
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4967 <p xml:lang="lat"> |
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4968 <s xml:id="echoid-s462" xml:space="preserve"> |
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|
4969 parabola |
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4970 </s> |
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|
4971 </p> |
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4972 <pb file="add_6787_f358v" o="358v" n="715"/> |
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|
4973 <div xml:id="echoid-div102" type="page_commentary" level="2" n="102"> |
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4974 <p> |
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4975 <s xml:id="echoid-s463" xml:space="preserve">[<emph style="it">Note: |
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4976 <p> |
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4977 <s xml:id="echoid-s463" xml:space="preserve"> |
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|
4978 This page refers to Propositions II.1, II.2, and I.21 from Apollonius, |
|
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|
4979 <emph style="it">Conicorum libri quattuor</emph>. |
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|
4980 </s> |
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4981 <lb/> |
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|
4982 <quote> |
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|
4983 II.1 |
|
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|
4984 If a straight line touch an hyperbola at its vertex, and from it on both sides of the diamter |
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|
4985 a straight line is cut off equal in square to the fourth of the figure, |
|
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|
4986 then the straight lines drawn from the centre of the section to the ends thus taken on the tangent |
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|
4987 will not meet the section. |
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|
4988 </quote> |
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|
4989 <lb/> |
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|
4990 <quote> |
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|
4991 II.2 |
|
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|
4992 With the same things it is to be shown that a straight line cutting the angle contained by |
|
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|
4993 the straight lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>E</mi></mstyle></math> is not another asymptote. |
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|
4994 </quote> |
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|
4995 <lb/> |
|
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|
4996 <quote> |
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|
4997 I.21 |
|
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|
4998 If in a hyperbola or ellipse or circumference of a circle straight lines are dropped as ordinates to the diameter, |
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|
4999 the square on them will be to the areas contained by the straight lines cut off by them beginning from the ends of |
|
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|
5000 the transverse side of the figure, as the upright side of the figure is to the transverse, |
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|
5001 and to each other as the areas contained by the straight lines cut off, as we have said. |
|
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|
5002 </quote> |
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|
5003 </p> |
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|
5004 </emph>] |
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|
5005 <lb/><lb/></s></p></div> |
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|
5006 <p xml:lang="lat"> |
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|
5007 <s xml:id="echoid-s465" xml:space="preserve"> |
|
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|
5008 prop. 2 |
|
22d6a63640c6
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|
5009 <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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|
5010 Proposition 2. |
|
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|
5011 </emph>]<lb/> |
|
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|
5012 </s> |
|
22d6a63640c6
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|
5013 </p> |
|
22d6a63640c6
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|
5014 <p xml:lang="lat"> |
|
22d6a63640c6
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|
5015 <s xml:id="echoid-s466" xml:space="preserve"> |
|
22d6a63640c6
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|
5016 lib. 2. prop: 1. |
|
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|
5017 <lb/>[<emph style="it">tr: |
|
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|
5018 Book 2, Proposition 1. |
|
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|
5019 </emph>]<lb/> |
|
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|
5020 </s> |
|
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|
5021 </p> |
|
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|
5022 <p xml:lang="lat"> |
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|
5023 <s xml:id="echoid-s467" xml:space="preserve"> |
|
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|
5024 per. 21 |
|
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|
5025 <lb/>[<emph style="it">tr: |
|
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|
5026 by Proposition 21. |
|
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|
5027 </emph>]<lb/> |
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|
5028 </s> |
|
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|
5029 <lb/> |
|
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5030 <s xml:id="echoid-s468" xml:space="preserve"> |
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|
5031 absurdum |
|
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|
5032 <lb/>[<emph style="it">tr: |
|
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|
5033 absurd |
|
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|
5034 </emph>]<lb/> |
|
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|
5035 </s> |
|
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|
5036 </p> |
|
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|
5037 <p xml:lang="lat"> |
|
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|
5038 <s xml:id="echoid-s469" xml:space="preserve"> |
|
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|
5039 absurdum |
|
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|
5040 <lb/>[<emph style="it">tr: |
|
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|
5041 absurd |
|
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|
5042 </emph>]<lb/> |
|
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|
5043 </s> |
|
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|
5044 </p> |
|
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|
5045 <p xml:lang="lat"> |
|
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|
5046 <s xml:id="echoid-s470" xml:space="preserve"> |
|
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|
5047 prop. 2 |
|
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|
5048 <lb/>[<emph style="it">tr: |
|
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|
5049 Proposition 2. |
|
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|
5050 </emph>]<lb/> |
|
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|
5051 </s> |
|
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|
5052 </p> |
|
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|
5053 <p xml:lang="lat"> |
|
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|
5054 <s xml:id="echoid-s471" xml:space="preserve"> |
|
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|
5055 Ergo absurdum |
|
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|
5056 <lb/>[<emph style="it">tr: |
|
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|
5057 Therefore absurd |
|
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|
5058 </emph>]<lb/> |
|
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|
5059 </s> |
|
22d6a63640c6
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|
5060 </p> |
|
22d6a63640c6
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|
5061 <pb file="add_6787_f359" o="359" n="716"/> |
|
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|
5062 <div xml:id="echoid-div103" type="page_commentary" level="2" n="103"> |
|
22d6a63640c6
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|
5063 <p> |
|
22d6a63640c6
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|
5064 <s xml:id="echoid-s472" xml:space="preserve">[<emph style="it">Note: |
|
22d6a63640c6
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|
5065 <p> |
|
22d6a63640c6
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|
5066 <s xml:id="echoid-s472" xml:space="preserve"> |
|
22d6a63640c6
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|
5067 The inclusion of a page number confirms that Harriot was using Commandino's edition of Apollonius, |
|
22d6a63640c6
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|
5068 <emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566). |
|
22d6a63640c6
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|
5069 Proposition 30 is a property of the ellipse. |
|
22d6a63640c6
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|
5070 </s> |
|
22d6a63640c6
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|
5071 <lb/> |
|
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|
5072 <quote> |
|
22d6a63640c6
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diff
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|
5073 I.30 |
|
22d6a63640c6
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|
5074 If in an ellipse or in opposite sections a straight line is drawn in both directions from the centre, |
|
22d6a63640c6
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|
5075 meeting the section, it will be bisected at the centre. |
|
22d6a63640c6
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|
5076 </quote> |
|
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|
5077 </p> |
|
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|
5078 </emph>] |
|
22d6a63640c6
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|
5079 <lb/><lb/></s></p></div> |
|
22d6a63640c6
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|
5080 <head xml:id="echoid-head102" xml:space="preserve" xml:lang="lat"> |
|
22d6a63640c6
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|
5081 Appol. 22.b. <lb/> |
|
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|
5082 lib. 1. pro. 30. |
|
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|
5083 <lb/>[<emph style="it">tr: |
|
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diff
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|
5084 Apollonius, page 22v, Book I, Proposition 30. |
|
22d6a63640c6
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|
5085 </emph>]<lb/> |
|
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|
5086 </head> |
|
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|
5087 <p xml:lang="lat"> |
|
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|
5088 <s xml:id="echoid-s474" xml:space="preserve"> |
|
22d6a63640c6
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|
5089 In elipsij |
|
22d6a63640c6
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|
5090 <lb/>[<emph style="it">tr: |
|
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|
5091 In an ellipse |
|
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|
5092 </emph>]<lb/> |
|
22d6a63640c6
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|
5093 </s> |
|
22d6a63640c6
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|
5094 </p> |
|
22d6a63640c6
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|
5095 <pb file="add_6787_f359v" o="359v" n="717"/> |
|
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|
5096 <pb file="add_6787_f360" o="360" n="718"/> |
|
22d6a63640c6
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|
5097 <div xml:id="echoid-div104" type="page_commentary" level="2" n="104"> |
|
22d6a63640c6
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5098 <p> |
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5099 <s xml:id="echoid-s475" xml:space="preserve">[<emph style="it">Note: |
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5100 <p> |
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5101 <s xml:id="echoid-s475" xml:space="preserve"> |
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5102 The inclusion of a page number confirms that Harriot was using Commandino's edition of Apollonius, |
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5103 <emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566). |
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|
5104 Proposition 33 explains how to find the tangent to any point of a parabola. |
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5105 </s> |
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5106 <lb/> |
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5107 <quote> |
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|
5108 I.33 |
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5109 If on a parabola some point is taken, and from it an ordinate is dropped to the diameter, and, |
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5110 to the straight line cut off by it on the diameter from the vertex, |
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5111 a straight line in the same straight line from its extremity is made equal, |
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5112 then the straight line joined from the point thus resulting to the point taken will touch the section. |
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5113 </quote> |
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5114 </p> |
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5115 </emph>] |
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5116 <lb/><lb/></s></p></div> |
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5117 <head xml:id="echoid-head103" xml:space="preserve" xml:lang="lat"> |
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5118 Appoll. pa: 24.b <lb/> |
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5119 lib. 1. pr. 33 |
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5120 <lb/>[<emph style="it">tr: |
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|
5121 Apollonius, page 24v, Book I, Proposition 33. |
|
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5122 </emph>]<lb/> |
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5123 </head> |
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5124 <pb file="add_6787_f360v" o="360v" n="719"/> |
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5125 <pb file="add_6787_f361" o="361" n="720"/> |
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5126 <div xml:id="echoid-div105" type="page_commentary" level="2" n="105"> |
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|
5127 <p> |
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5128 <s xml:id="echoid-s477" xml:space="preserve">[<emph style="it">Note: |
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|
5129 <p> |
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5130 <s xml:id="echoid-s477" xml:space="preserve"> |
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|
5131 The inclusion of a page number confirms that Harriot was using Commandino's edition of Apollonius, |
|
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5132 <emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566). |
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|
5133 Proposition 37 gives a property of a tangent to a hyperbola or ellipse or circle. |
|
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|
5134 </s> |
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|
5135 <lb/> |
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|
5136 <quote> |
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|
5137 I.37 |
|
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|
5138 If a straight line touching an hyperbola or ellipse or circumference of a circle meets the diameter, |
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|
5139 and from the point of contact to the diameter a straight line is dropped as ordinate, |
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|
5140 then the straight line cut off by the ordinate from the centre of the section |
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|
5141 with the straight line cut off by the tangent from the centre of the section will contain |
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|
5142 an area equal to the square on the radius of the section, |
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|
5143 and with the straight line between the ordinate and the tangent will contain an area |
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|
5144 having the ratio to the square on the ordinate which the transverse has to the upright. |
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|
5145 </quote> |
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|
5146 </p> |
|
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|
5147 </emph>] |
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|
5148 <lb/><lb/></s></p></div> |
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|
5149 <head xml:id="echoid-head104" xml:space="preserve" xml:lang="lat"> |
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|
5150 lib. 1. <lb/> |
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|
5151 Appol. pag. 27. <lb/> |
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|
5152 prop. 37. |
|
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|
5153 <lb/>[<emph style="it">tr: |
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|
5154 Book I, Apollonius, page 27, Proposition 37. |
|
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|
5155 </emph>]<lb/> |
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|
5156 </head> |
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|
5157 <p xml:lang="lat"> |
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|
5158 <s xml:id="echoid-s479" xml:space="preserve"> |
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|
5159 Dico quod |
|
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|
5160 <lb/>[<emph style="it">tr: |
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|
5161 I say that |
|
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|
5162 </emph>]<lb/> |
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|
5163 </s> |
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|
5164 </p> |
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|
5165 <p xml:lang="lat"> |
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|
5166 <s xml:id="echoid-s480" xml:space="preserve"> |
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|
5167 In hyperbola |
|
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|
5168 <lb/>[<emph style="it">tr: |
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|
5169 In a hyerbola |
|
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|
5170 </emph>]<lb/> |
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|
5171 </s> |
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|
5172 </p> |
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|
5173 <p xml:lang="lat"> |
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5174 <s xml:id="echoid-s481" xml:space="preserve"> |
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|
5175 In elipsi & circulo |
|
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|
5176 <lb/>[<emph style="it">tr: |
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|
5177 In an ellipse and circle |
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|
5178 </emph>]<lb/> |
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|
5179 </s> |
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|
5180 </p> |
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5181 <pb file="add_6787_f361v" o="361v" n="721"/> |
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|
5182 <pb file="add_6787_f362" o="362" n="722"/> |
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|
5183 <div xml:id="echoid-div106" type="page_commentary" level="2" n="106"> |
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|
5184 <p> |
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5185 <s xml:id="echoid-s482" xml:space="preserve">[<emph style="it">Note: |
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5186 <p> |
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5187 <s xml:id="echoid-s482" xml:space="preserve"> |
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5188 The inclusion of a page number confirms that Harriot was using Commandino's edition of Apollonius, |
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5189 <emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566). |
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5190 Proposition 38 gives a further property of a tangent to a hyperbola or ellipse or circle |
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5191 (see also Add MS 6787, f. 361). |
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5192 </s> |
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5193 <lb/> |
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5194 <quote> |
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5195 I.38 |
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5196 If a straight line touching a hyperbola or ellipse or circumference of a circle meets the second diameter, |
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5197 and if from the point of contact a straight line is dropped to that same [second] diameter parallel |
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5198 to the other diameter, then the straight line cut off from the centre of the section |
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5199 by the dropped straight line, together with the straight line cut off [on the second diameter] |
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5200 by the tangent from the centre of the section will contain an area |
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5201 equal to the square on the half of the second diameter, and, |
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5202 together with the straight line [on the second diameter] between the dropped straight line and the tangent, |
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5203 will contain an area having a ratio to the square on the dropped straight line |
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5204 which the upright side of the figure has to the transverse. |
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5205 </quote> |
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5206 </p> |
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5207 </emph>] |
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5208 <lb/><lb/></s></p></div> |
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5209 <head xml:id="echoid-head105" xml:space="preserve" xml:lang="lat"> |
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5210 Appol. pag. 28 <lb/> |
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5211 pro: 38. |
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5212 <lb/>[<emph style="it">tr: |
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5213 Apollonius, page 28, Proposition 38. |
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5214 </emph>]<lb/> |
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5215 </head> |
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5216 <p xml:lang="lat"> |
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5217 <s xml:id="echoid-s484" xml:space="preserve"> |
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5218 Iisdem positis. |
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5219 <lb/>[<emph style="it">tr: |
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5220 Under the same suppositions. |
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5221 </emph>]<lb/> |
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5222 </s> |
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5223 <lb/> |
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5224 <s xml:id="echoid-s485" xml:space="preserve"> |
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5225 1. casus: hyperboles |
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5226 <lb/>[<emph style="it">tr: |
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5227 Case 1. Hyperbolas |
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5228 </emph>]<lb/> |
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5229 </s> |
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5230 </p> |
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5231 <p xml:lang="lat"> |
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5232 <s xml:id="echoid-s486" xml:space="preserve"> |
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5233 Sed: <lb/> |
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5234 in elipsi <lb/> |
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5235 per 36 |
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5236 <lb/>[<emph style="it">tr: |
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5237 But in an ellipse, by proposition 36 |
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5238 </emph>]<lb/> |
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5239 </s> |
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5240 </p> |
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5241 <p xml:lang="lat"> |
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5242 <s xml:id="echoid-s487" xml:space="preserve"> |
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5243 In Ellipsi |
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5244 <lb/>[<emph style="it">tr: |
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5245 In an ellipse |
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5246 </emph>]<lb/> |
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5247 </s> |
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5248 </p> |
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5249 <pb file="add_6787_f362v" o="362v" n="723"/> |
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5250 <div xml:id="echoid-div107" type="page_commentary" level="2" n="107"> |
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5251 <p> |
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5252 <s xml:id="echoid-s488" xml:space="preserve">[<emph style="it">Note: |
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5253 <p> |
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5254 <s xml:id="echoid-s488" xml:space="preserve"> |
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5255 This page refers to Propositions 21, 36, and 37 from Book I of Apollonius, as edited by Commandino in |
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5256 <emph style="it">Conicorum libri quattuor</emph> (1566). |
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5257 </s> |
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5258 <lb/> |
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5259 <quote> |
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|
5260 I.21. |
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|
5261 If in a hyperbola or ellipse or circumference of a circle straight lines |
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5262 are dropped as ordinates to the diameter, the square on them will be to the areas contained |
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5263 by the straight lines cut off by them beginning from the ends of the transverse side of the figure, |
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5264 as the upright side of the figure is to the transverse, and to each other as the areas contained |
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5265 by the straight lines cut off, as we have said. |
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5266 </quote> |
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5267 <lb/> |
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|
5268 <quote> |
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|
5269 I.36. |
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|
5270 If some straight line, meeting the transverse side of the figure touches an hyperbola |
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5271 or ellipse or circumference of a circle, and if a straight line is dropped from the point of contact |
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5272 as an ordinate to the diameter, then as the straight line cut off by the tangent |
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|
5273 from the end of the transverse side is to the straight line cut off by the tangent from the other end of that side, |
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5274 so will the straight line cut off by the ordinate from the end of the side be |
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|
5275 to the straight line cut off by the ordinate from the other end of the side in such a way that |
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5276 the corresponding straight lines are continuous; and another straight line will not fall into the space |
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|
5277 between the tangent and the section of the cone. |
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|
5278 </quote> |
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|
5279 <lb/> |
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|
5280 <quote> |
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|
5281 I.37 |
|
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|
5282 If a straight line touching an hyperbola or ellipse or circumference of a circle meets the diameter, |
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|
5283 and from the point of contact to the diameter a straight line is dropped as ordinate, |
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|
5284 then the straight line cut off by the ordinate from the centre of the section |
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|
5285 with the straight line cut off by the tangent from the centre of the section will contain |
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|
5286 an area equal to the square on the radius of the section, |
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|
5287 and with the straight line between the ordinate and the tangent will contain an area |
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|
5288 having the ratio to the square on the ordinate which the transverse has to the upright. |
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|
5289 </quote> |
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|
5290 </p> |
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|
5291 </emph>] |
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|
5292 <lb/><lb/></s></p></div> |
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|
5293 <p xml:lang="lat"> |
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5294 <s xml:id="echoid-s490" xml:space="preserve"> |
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|
5295 Nota |
|
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5296 <lb/>[<emph style="it">tr: |
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|
5297 Note |
|
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|
5298 </emph>]<lb/> |
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|
5299 </s> |
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|
5300 <lb/> |
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|
5301 <s xml:id="echoid-s491" xml:space="preserve"> |
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|
5302 per 37: |
|
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|
5303 <lb/>[<emph style="it">tr: |
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|
5304 by proposition 37 |
|
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|
5305 </emph>]<lb/> |
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|
5306 </s> |
|
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|
5307 <lb/> |
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5308 <s xml:id="echoid-s492" xml:space="preserve"> |
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|
5309 per 21: |
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|
5310 <lb/>[<emph style="it">tr: |
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|
5311 by proposition 21 |
|
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|
5312 </emph>]<lb/> |
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|
5313 </s> |
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|
5314 <lb/> |
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5315 <s xml:id="echoid-s493" xml:space="preserve"> |
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|
5316 per, 36: ita: |
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5317 <lb/>[<emph style="it">tr: |
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|
5318 by proposition 36, thus: |
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|
5319 </emph>]<lb/> |
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|
5320 </s> |
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|
5321 <lb/> |
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5322 <s xml:id="echoid-s494" xml:space="preserve"> |
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|
5323 igitur: ita: |
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|
5324 <lb/>[<emph style="it">tr: |
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|
5325 therefore, thus: |
|
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|
5326 </emph>]<lb/> |
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|
5327 </s> |
|
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|
5328 </p> |
|
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|
5329 <pb file="add_6787_f363" o="363" n="724"/> |
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|
5330 <div xml:id="echoid-div108" type="page_commentary" level="2" n="108"> |
|
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|
5331 <p> |
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|
5332 <s xml:id="echoid-s495" xml:space="preserve">[<emph style="it">Note: |
|
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|
5333 <p> |
|
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|
5334 <s xml:id="echoid-s495" xml:space="preserve"> |
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|
5335 For proposition 38, see Add MS 6787, f. 362. |
|
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|
5336 </s> |
|
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|
5337 </p> |
|
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|
5338 </emph>] |
|
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|
5339 <lb/><lb/></s></p></div> |
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|
5340 <head xml:id="echoid-head106" xml:space="preserve" xml:lang="lat"> |
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|
5341 Appol: lib. 1. pag. 28. <lb/> |
|
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|
5342 prop: 38 |
|
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|
5343 <lb/>[<emph style="it">tr: |
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|
5344 Apollonius, page 28, Proposition 38. |
|
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|
5345 </emph>]<lb/> |
|
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|
5346 </head> |
|
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|
5347 <p xml:lang="lat"> |
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|
5348 <s xml:id="echoid-s497" xml:space="preserve"> |
|
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|
5349 Dico quod: |
|
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|
5350 <lb/>[<emph style="it">tr: |
|
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|
5351 I say that |
|
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|
5352 </emph>]<lb/> |
|
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|
5353 </s> |
|
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|
5354 </p> |
|
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|
5355 <p xml:lang="lat"> |
|
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|
5356 <s xml:id="echoid-s498" xml:space="preserve"> |
|
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|
5357 2. Casus |
|
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|
5358 <lb/>[<emph style="it">tr: |
|
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|
5359 Case 2. |
|
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|
5360 </emph>]<lb/> |
|
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|
5361 </s> |
|
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|
5362 </p> |
|
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|
5363 <pb file="add_6787_f363v" o="363v" n="725"/> |
|
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|
5364 <pb file="add_6787_f364" o="364" n="726"/> |
|
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|
5365 <div xml:id="echoid-div109" type="page_commentary" level="2" n="109"> |
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5366 <p> |
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5367 <s xml:id="echoid-s499" xml:space="preserve">[<emph style="it">Note: |
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5368 <p> |
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5369 <s xml:id="echoid-s499" xml:space="preserve"> |
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5370 The inclusion of a page number confirms that Harriot was using Commandino's edition of Apollonius, |
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5371 <emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566). |
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5372 Proposition 39 gives a further property of a tangent to a hyperbola or ellipse or circle |
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5373 (see also Add MS 6787, f. 361 and f. 362). |
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5374 </s> |
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5375 <lb/> |
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5376 <quote> |
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5377 I.39 |
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5378 If a straight line touching a hyperbola or ellipse or circumference of a circle meets the diameter, |
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5379 and if from the point of contact a straight line is dropped as ordinate to the diameter, |
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5380 then whichever of the two straight lines is taken, of which one is the straight line between the |
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5381 [intersection of the] ordinate [with the diameter] and the centre of the section, |
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5382 and the other is between [the intersection of] the ordinate and the tangent [with the diameter], |
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5383 the ordinate will have to it the ratio compounded of the ratio of the other of the two straight lines |
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5384 to the ordinate and of the ratio of the upright side of the figure to the transverse. |
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5385 </quote> |
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5386 </p> |
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5387 </emph>] |
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5388 <lb/><lb/></s></p></div> |
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5389 <head xml:id="echoid-head107" xml:space="preserve" xml:lang="lat"> |
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5390 lib. 1. <lb/> |
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|
5391 App. pag. 29. <lb/> |
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5392 pro: 39. |
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5393 <lb/>[<emph style="it">tr: |
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|
5394 Book I, Apollonius, page 29, Proposition 39. |
|
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5395 </emph>]<lb/> |
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5396 </head> |
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5397 <p xml:lang="lat"> |
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5398 <s xml:id="echoid-s501" xml:space="preserve"> |
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|
5399 Nostro modo <lb/> |
|
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|
5400 brevissime. |
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5401 <lb/>[<emph style="it">tr: |
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|
5402 My method, very brief. |
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5403 </emph>]<lb/> |
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5404 </s> |
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5405 </p> |
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5406 <p xml:lang="lat"> |
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5407 <s xml:id="echoid-s502" xml:space="preserve"> |
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5408 Sed: <lb/> |
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|
5409 in elipsi <lb/> |
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|
5410 per 36 |
|
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5411 <lb/>[<emph style="it">tr: |
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5412 But in an ellipse, by proposition 36 |
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5413 </emph>]<lb/> |
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5414 </s> |
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|
5415 </p> |
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5416 <p xml:lang="lat"> |
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5417 <s xml:id="echoid-s503" xml:space="preserve"> |
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|
5418 In Ellipsi |
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5419 <lb/>[<emph style="it">tr: |
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|
5420 In an ellipse |
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5421 </emph>]<lb/> |
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5422 </s> |
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5423 </p> |
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|
5424 <pb file="add_6787_f364v" o="364v" n="727"/> |
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5425 <div xml:id="echoid-div110" type="page_commentary" level="2" n="110"> |
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5426 <p> |
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5427 <s xml:id="echoid-s504" xml:space="preserve">[<emph style="it">Note: |
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5428 <p> |
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5429 <s xml:id="echoid-s504" xml:space="preserve"> |
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|
5430 This page refers to Propositions I.38 and I.40 from Apollonius, |
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|
5431 <emph style="it">Conicorum libri quattuor</emph>. |
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5432 </s> |
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|
5433 <lb/> |
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5434 <quote> |
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|
5435 I.38 |
|
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|
5436 If a straight line touching a hyperbola or ellipse or circumference of a circle meets the second diameter, |
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|
5437 and if from the point of contact a straight line is dropped to that same [second] diameter parallel |
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|
5438 to the other diameter, then the straight line cut off from the centre of the section |
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|
5439 by the dropped straight line, together with the straight line cut off [on the second diameter] |
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5440 by the tangent from the centre of the section will contain an area |
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|
5441 equal to the square on the half of the second diameter, and, |
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|
5442 together with the straight line [on the second diameter] between the dropped straight line and the tangent, |
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|
5443 will contain an area having a ratio to the square on the dropped straight line |
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5444 which the upright side of the figure has to the transverse. |
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|
5445 </quote> |
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|
5446 <lb/> |
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|
5447 <quote> |
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|
5448 I.40. |
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5449 If a straight line touching a hyperbola or ellipse or circumference of a circle meets the second diameter, |
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|
5450 and if from the point of contact a straight line is dropped to the same diameter parallel to the other diameter, |
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|
5451 then whichever of the two straight lines is taken [along the second diameter], of which one is |
|
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|
5452 the straight line between the dropped straight line and the centre of the section, and the other is |
|
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|
5453 between the dropped straight line and the tangent, the dropped straight line will have to it |
|
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5454 the ratio compounded of the ratio of the transverse side to the upright and of |
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5455 the ratio of the other of the two straight lines to the dropped straight line. |
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5456 </quote> |
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5457 </p> |
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5458 </emph>] |
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5459 <lb/><lb/></s></p></div> |
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5460 <p xml:lang="lat"> |
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5461 <s xml:id="echoid-s506" xml:space="preserve"> |
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5462 prop: 40 |
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5463 <lb/>[<emph style="it">tr: |
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5464 proposition 40 |
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5465 </emph>]<lb/> |
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5466 </s> |
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5467 <lb/> |
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5468 <s xml:id="echoid-s507" xml:space="preserve"> |
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5469 per 38: |
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5470 <lb/>[<emph style="it">tr: |
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5471 by proposition 38 |
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5472 </emph>]<lb/> |
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5473 </s> |
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5474 </p> |
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5475 <pb file="add_6787_f365" o="365" n="728"/> |
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5476 <div xml:id="echoid-div111" type="page_commentary" level="2" n="111"> |
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5477 <p> |
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5478 <s xml:id="echoid-s508" xml:space="preserve">[<emph style="it">Note: |
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5479 <p> |
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5480 <s xml:id="echoid-s508" xml:space="preserve"> |
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5481 The inclusion of a page number confirms that Harriot was using Commandino's edition of Apollonius, |
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5482 <emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566). |
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5483 </s> |
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5484 <lb/> |
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5485 <quote> |
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5486 I.41 |
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5487 If in a hyperbola or ellipse or circumference of a circle a straight line is dropped as ordinate to the diameter, |
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5488 and if equiangular parallelograms are described both on the ordinate and on the radius, |
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5489 and if the ordinate side has to the remaining side of the figure the ratio compounded of the ratio of the radius |
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5490 to the remaining side of its figure, and the ratio of the upright side of the section 19s figure to the transverse, |
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5491 then the figure on the straight line between the centre and the ordinate, similar to the figure on the radius, |
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5492 is in the case of the hyperbola greater than the figure on the ordinate by the figure on the radius, and, |
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5493 in the case of the ellipse and circumference of a circle, together with the figure on the ordinate |
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5494 is equal to the figure on the radius. |
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5495 </quote> |
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5496 </p> |
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5497 </emph>] |
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5498 <lb/><lb/></s></p></div> |
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5499 <head xml:id="echoid-head108" xml:space="preserve" xml:lang="lat"> |
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5500 pag: 29. b. <lb/> |
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5501 Appol. pro: 41 |
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5502 <lb/>[<emph style="it">tr: |
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|
5503 page 29v, Apollonius, Proposition 41. |
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5504 </emph>]<lb/> |
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5505 </head> |
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5506 <p xml:lang="lat"> |
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5507 <s xml:id="echoid-s510" xml:space="preserve"> |
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5508 Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> ordinata <lb/> |
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|
5509 et parallelogramma <lb/> |
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5510 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>g</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> æquiangula |
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5511 <lb/>[<emph style="it">tr: |
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5512 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> be an ordinate, and parallelograms <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>g</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> equiangular. |
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5513 </emph>]<lb/> |
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5514 </s> |
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5515 <lb/> |
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5516 <s xml:id="echoid-s511" xml:space="preserve"> |
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5517 composita ex |
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5518 <lb/>[<emph style="it">tr: |
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|
5519 composed from |
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5520 </emph>]<lb/> |
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5521 </s> |
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5522 </p> |
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5523 <p xml:lang="lat"> |
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5524 <s xml:id="echoid-s512" xml:space="preserve"> |
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|
5525 Dico quod: in Hyperbole |
|
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5526 <lb/>[<emph style="it">tr: |
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|
5527 I say that, in a hyperbola |
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5528 </emph>]<lb/> |
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|
5529 </s> |
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5530 <lb/> |
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5531 <s xml:id="echoid-s513" xml:space="preserve"> |
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|
5532 In ellipsi et circulo |
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5533 <lb/>[<emph style="it">tr: |
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|
5534 In an ellipse and circle |
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|
5535 </emph>]<lb/> |
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|
5536 </s> |
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|
5537 </p> |
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|
5538 <pb file="add_6787_f365v" o="365v" n="729"/> |
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5539 <pb file="add_6787_f366" o="366" n="730"/> |
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|
5540 <div xml:id="echoid-div112" type="page_commentary" level="2" n="112"> |
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5541 <p> |
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5542 <s xml:id="echoid-s514" xml:space="preserve">[<emph style="it">Note: |
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5543 <p> |
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5544 <s xml:id="echoid-s514" xml:space="preserve"> |
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5545 The inclusion of a page number confirms that Harriot was using Commandino's edition of Apollonius, |
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5546 <emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566). <lb/> |
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5547 For proposition 41, see Add MS 6787, f. 365. |
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5548 </s> |
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5549 </p> |
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5550 </emph>] |
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5551 <lb/><lb/></s></p></div> |
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5552 <head xml:id="echoid-head109" xml:space="preserve" xml:lang="lat"> |
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5553 pag: 29. <lb/> |
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5554 Ap. pro: 41 |
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5555 <lb/>[<emph style="it">tr: |
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5556 page 29, Apollonius, Proposition 41. |
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5557 </emph>]<lb/> |
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5558 </head> |
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5559 <pb file="add_6787_f366v" o="366v" n="731"/> |
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5560 <pb file="add_6787_f367" o="367" n="732"/> |
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5561 <div xml:id="echoid-div113" type="page_commentary" level="2" n="113"> |
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5562 <p> |
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5563 <s xml:id="echoid-s516" xml:space="preserve">[<emph style="it">Note: |
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5564 <p> |
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5565 <s xml:id="echoid-s516" xml:space="preserve"> |
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5566 The inclusion of a page number confirms that Harriot was using Commandino's edition of Apollonius, |
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5567 <emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566). <lb/> |
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5568 For Proposition 11, the original definition of a parabola, see Add MS 6787, f. 357. |
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5569 </s> |
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5570 <lb/> |
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5571 <quote> |
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5572 I. 52 |
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5573 Given a straight line in a plane bounded at one point, to find in the plane the section of a cone called parabola, |
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5574 whose diameter is the given straight line, and whose vertex is the end of the straight line, |
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5575 and where whatever straight line is dropped from the section to the diameter at a given angle, |
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5576 will equal in square the rectangle contained by the straight line cut off by it from the vertex of the section |
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5577 and by some other given straight line. |
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5578 </quote> |
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5579 </p> |
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5580 </emph>] |
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5581 <lb/><lb/></s></p></div> |
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5582 <head xml:id="echoid-head110" xml:space="preserve" xml:lang="lat"> |
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5583 pag. 37. <lb/> |
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5584 Appol. pro: 52 |
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5585 <lb/>[<emph style="it">tr: |
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5586 page 37, Apollonius, Proposition 52. |
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5587 </emph>]<lb/> |
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5588 </head> |
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5589 <p xml:lang="lat"> |
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5590 <s xml:id="echoid-s518" xml:space="preserve"> |
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5591 ad latus rectum |
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5592 <lb/>[<emph style="it">tr: |
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5593 for the latus rectum</emph>]<lb/> |
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5594 </s> |
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5595 </p> |
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5596 <p xml:lang="lat"> |
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5597 <s xml:id="echoid-s519" xml:space="preserve"> |
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5598 per. 11. ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>a</mi><mi>l</mi></mstyle></math> est sectio <lb/> |
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5599 cuius axis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> <lb/> |
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5600 et recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math>. |
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5601 <lb/>[<emph style="it">tr: |
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5602 by proposition 11, therefore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>a</mi><mi>l</mi></mstyle></math> is the section whose axis is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> with line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math>. |
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5603 </emph>]<lb/> |
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5604 </s> |
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5605 </p> |
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5606 <p xml:lang="lat"> |
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5607 <s xml:id="echoid-s520" xml:space="preserve"> |
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5608 sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> diameter <lb/> |
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5609 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> recta <lb/> |
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5610 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>a</mi><mi>e</mi></mstyle></math> angulus appl. <lb/> |
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5611 non recta |
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5612 <lb/>[<emph style="it">tr: |
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5613 let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> be the diameter, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> the line, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>a</mi><mi>e</mi></mstyle></math> the angle of application, not a right angle. |
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5614 </emph>]<lb/> |
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5615 </s> |
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5616 </p> |
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5617 <p xml:lang="lat"> |
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5618 <s xml:id="echoid-s521" xml:space="preserve"> |
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5619 unde fit sectio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi></mstyle></math> ex cono recto <lb/> |
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5620 ut supra. et transit per <lb/> |
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5621 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>a</mi></mstyle></math> est contingens, quia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>k</mi><mo>=</mo><mi>k</mi><mi>l</mi></mstyle></math>. |
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5622 <lb/>[<emph style="it">tr: |
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5623 whence arises the section <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi></mstyle></math> from the right cone as above, and the crossing line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>a</mi></mstyle></math> is a tangent, |
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5624 because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>k</mi><mo>=</mo><mi>k</mi><mi>l</mi></mstyle></math>. |
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5625 </emph>]<lb/> |
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5626 </s> |
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5627 </p> |
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5628 <p xml:lang="lat"> |
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5629 <s xml:id="echoid-s522" xml:space="preserve"> |
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5630 Ergo per 49 <lb/> |
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5631 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> est latus <lb/> |
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5632 rectum. et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> diameter &c. |
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5633 <lb/>[<emph style="it">tr: |
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5634 Therefore by pproposition 49, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> is the latus rectum and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> the diameter. |
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5635 </emph>]<lb/> |
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5636 </s> |
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5637 </p> |
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5638 <pb file="add_6787_f367v" o="367v" n="733"/> |
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5639 <pb file="add_6787_f368" o="368" n="734"/> |
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5640 <pb file="add_6787_f368v" o="368v" n="735"/> |
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5641 <pb file="add_6787_f369" o="369" n="736"/> |
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5642 <pb file="add_6787_f369v" o="369v" n="737"/> |
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5643 <pb file="add_6787_f370" o="370" n="738"/> |
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5644 <pb file="add_6787_f370v" o="370v" n="739"/> |
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5645 <pb file="add_6787_f371" o="371" n="740"/> |
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5646 <pb file="add_6787_f371v" o="371v" n="741"/> |
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5647 <pb file="add_6787_f372" o="372" n="742"/> |
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5648 <div xml:id="echoid-div114" type="page_commentary" level="2" n="114"> |
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5649 <p> |
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5650 <s xml:id="echoid-s523" xml:space="preserve">[<emph style="it">Note: |
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5651 <p> |
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5652 <s xml:id="echoid-s523" xml:space="preserve"> |
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5653 This page refers to Proposition 21 of Book I of Apollonius, as edited by Commandino in |
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5654 <emph style="it">Conicorum libri quattuor</emph> (1566). |
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5655 </s> |
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5656 <lb/> |
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5657 <quote> |
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5658 I.21 |
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5659 If in a hyperbola or ellipse or circumference of a circle straight lines are dropped as ordinates to the diameter, |
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5660 the square on them will be to the areas contained by the straight lines cut off by them beginning from the ends of |
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5661 the transverse side of the figure, as the upright side of the figure is to the transverse, |
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5662 and to each other as the areas contained by the straight lines cut off, as we have said. |
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5663 </quote> |
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5664 </p> |
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5665 </emph>] |
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5666 <lb/><lb/></s></p></div> |
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5667 <head xml:id="echoid-head111" xml:space="preserve" xml:lang="lat"> |
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5668 De ellipsi. |
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5669 <lb/>[<emph style="it">tr: |
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5670 On the ellipse |
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5671 </emph>]<lb/> |
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5672 </head> |
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5673 <p xml:lang="lat"> |
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5674 <s xml:id="echoid-s525" xml:space="preserve"> |
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5675 Theorema |
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5676 </s> |
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5677 <lb/> |
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5678 <s xml:id="echoid-s526" xml:space="preserve"> |
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5679 Si quotlibet lineis ordinatim applicatis ad diametrum circuli, aliæ numero <lb/> |
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5680 et quantitate æquales similiter applicentur ad similes partes lineæ maioris <emph style="st">vel</emph> <lb/> |
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5681 minoris <emph style="super">de</emph> data diametro circuli: termini illarum linearum sunt in ellipsi. |
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5682 <lb/>[<emph style="it">tr: |
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5683 If any number of ordinate lines are dropped to the diameter of a circle, that number of other lines, |
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5684 equal in quantity and similarly applied to similar parts of lines greater or less than the given diameter of a circe, |
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5685 then the ends of those lines are on an ellipse. |
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5686 </emph>]<lb/> |
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5687 </s> |
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5688 <lb/> |
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5689 <s xml:id="echoid-s527" xml:space="preserve"> |
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5690 Sit diameter circuli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math>. lineæ ordinatim applicatæ <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>e</mi><mi>f</mi></mstyle></math>. sit etiam <lb/> |
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5691 linea maior quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mi>x</mi></mstyle></math> cui applicentur ad angulos rectos <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>α</mo><mo>β</mo></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>ε</mo><mo>θ</mo></mstyle></math> æquales <lb/> |
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5692 <emph style="super">lineis</emph> <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>e</mi><mi>f</mi></mstyle></math> circuli. |
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5693 <emph style="st">Et sint partes lineæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mi>x</mi></mstyle></math> videlicet [???] quotlibet </emph> <lb/> |
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5694 <emph style="st">[???]</emph> <emph style="super">sed ita</emph> |
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5695 fiat ud <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mi>x</mi></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> ita <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mo>β</mo></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mo>θ</mo></mstyle></math> ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>f</mi></mstyle></math>. Quod etiam fit <lb/> |
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5696 si utraque lineæ <emph style="super"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mi>x</mi></mstyle></math></emph> similter dividantur et ad utraque partes |
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5697 <emph style="st">æquales</emph> similes et <lb/> |
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5698 similiter sitas æquales lineæ ordinatim applicentur. <lb/> |
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5699 Dico quod puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>α</mo></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>ε</mo></mstyle></math> termini linearum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>α</mo><mo>θ</mo></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>ε</mo><mo>θ</mo></mstyle></math> sunt in ellipsi. <lb/> |
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5700 Quoniam ex hypothesi <lb/> |
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5701 <lb/>[...]<lb/> |
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5702 Ergo: per 21, prop: primi Apollonij, puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>α</mo></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>ε</mo></mstyle></math> <lb/> |
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5703 sunt in ellipsi. quod demonstrare oportuit. |
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5704 <lb/>[<emph style="it">tr: |
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5705 Let the diameter of the circle be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math>, and the ordinate lines <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>e</mi><mi>f</mi></mstyle></math>; |
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5706 also let the line greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mi>x</mi></mstyle></math>, to which are applied at right angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>α</mo><mo>β</mo></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>ε</mo><mo>θ</mo></mstyle></math> |
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5707 euqal to the lines <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>e</mi><mi>f</mi></mstyle></math> in the circle. |
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5708 But thus, as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mi>x</mi></mstyle></math> is to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> so is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mo>β</mo></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mo>θ</mo></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>f</mi></mstyle></math>. |
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5709 Whcih also happens if the two lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>δ</mo><mi>x</mi></mstyle></math> are similarly divided and to both parts, |
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5710 similar and similarly situated, equal ordinate lines are applied. <lb/> |
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5711 I say that the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>α</mo></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>ε</mo></mstyle></math>, the ends of the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>α</mo><mo>θ</mo></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>ε</mo><mo>θ</mo></mstyle></math>, are on an ellipse. <lb/> |
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5712 Because from the hypothesis: <lb/> |
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5713 <lb/>[...]<lb/> |
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5714 Therefore, by Proposition 21 of the first Book of Apollonius, the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>α</mo></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>ε</mo></mstyle></math> are on an ellipse; |
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5715 which was to be demonstrated. |
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5716 </emph>]<lb/> |
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5717 </s> |
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5718 </p> |
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5719 <pb file="add_6787_f372v" o="372v" n="743"/> |
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5720 <pb file="add_6787_f373" o="373" n="744"/> |
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5721 <pb file="add_6787_f373v" o="373v" n="745"/> |
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5722 <pb file="add_6787_f374" o="374" n="746"/> |
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5723 <pb file="add_6787_f374v" o="374v" n="747"/> |
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5724 <pb file="add_6787_f375" o="375" n="748"/> |
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5725 <pb file="add_6787_f375v" o="375v" n="749"/> |
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5726 <pb file="add_6787_f376" o="376" n="750"/> |
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5727 <pb file="add_6787_f376v" o="376v" n="751"/> |
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5728 <pb file="add_6787_f377" o="377" n="752"/> |
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5729 <pb file="add_6787_f377v" o="377v" n="753"/> |
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5730 <p> |
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5731 <s xml:id="echoid-s528" xml:space="preserve"> |
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5732 Amen <lb/> |
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5733 Alexander King of M. <lb/> |
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5734 </s> |
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5735 </p> |
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5736 <p> |
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5737 <s xml:id="echoid-s529" xml:space="preserve"> |
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5738 [tomas haryots] <lb/> |
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5739 [Tomas Haryots] <lb/> |
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5740 [yn the begeneng] |
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5741 </s> |
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5742 </p> |
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5743 <pb file="add_6787_f378" o="378" n="754"/> |
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5744 <pb file="add_6787_f378v" o="378v" n="755"/> |
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5745 <pb file="add_6787_f379" o="379" n="756"/> |
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5746 <pb file="add_6787_f379v" o="379v" n="757"/> |
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5747 <pb file="add_6787_f380" o="380" n="758"/> |
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5748 <pb file="add_6787_f380v" o="380v" n="759"/> |
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5749 <pb file="add_6787_f381" o="381" n="760"/> |
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5750 <pb file="add_6787_f381v" o="381v" n="761"/> |
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5751 <pb file="add_6787_f382" o="382" n="762"/> |
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5752 <pb file="add_6787_f382v" o="382v" n="763"/> |
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5753 <pb file="add_6787_f383" o="383" n="764"/> |
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5754 <pb file="add_6787_f383v" o="383v" n="765"/> |
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5755 <pb file="add_6787_f384" o="384" n="766"/> |
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5756 <pb file="add_6787_f384v" o="384v" n="767"/> |
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5757 <pb file="add_6787_f385" o="385" n="768"/> |
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5758 <pb file="add_6787_f385v" o="385v" n="769"/> |
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5759 <pb file="add_6787_f386" o="386" n="770"/> |
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5760 <pb file="add_6787_f386v" o="386v" n="771"/> |
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5761 <pb file="add_6787_f387" o="387" n="772"/> |
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5762 <pb file="add_6787_f387v" o="387v" n="773"/> |
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5763 <pb file="add_6787_f388" o="388" n="774"/> |
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5764 <pb file="add_6787_f388v" o="388v" n="775"/> |
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5765 <pb file="add_6787_f389" o="389" n="776"/> |
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5766 <div xml:id="echoid-div115" type="page_commentary" level="2" n="115"> |
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5767 <p> |
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5768 <s xml:id="echoid-s530" xml:space="preserve">[<emph style="it">Note: |
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5769 <p> |
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5770 <s xml:id="echoid-s530" xml:space="preserve"> |
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5771 The reference to Pappus is to Commandino's edition of Books III to VIII, |
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5772 <emph style="it">Mathematicae collecitones</emph> (1558). |
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5773 The proposition on pages 320v–321 is Proposition 14. |
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5774 </s> |
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5775 <lb/> |
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5776 <quote xml:lang="lat"> |
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5777 Problema X. Propositio XIV. <lb/> |
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5778 Facile autem est inuentis quibuscumque coniugationibus diametrorum ellipsis, |
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5779 axes cuius organice inuenire. quod quidem hac ratione fiet. |
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5780 </quote> |
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5781 <lb/> |
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5782 <quote> |
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5783 Moreover, having found any conjugates of the diamters of the ellipse, |
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5784 it is easy to find the axes mechancially, which indeed are in this ratio. |
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5785 </quote> |
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5786 </p> |
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5787 </emph>] |
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5788 <lb/><lb/></s></p></div> |
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5789 <head xml:id="echoid-head112" xml:space="preserve" xml:lang="lat"> |
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5790 pappus. 321 <lb/> |
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5791 Datæ, coniugatæ elypseos <lb/> |
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5792 diametri <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>c</mi><mi>d</mi></mstyle></math>. <lb/> |
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5793 Quæruntur axes. |
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5794 <lb/>[<emph style="it">tr: |
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5795 Pappus, page 321. <lb/> |
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5796 Given conjugate ellipses with diameters <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>c</mi><mi>d</mi></mstyle></math>, there are sought the axes. |
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5797 </emph>]<lb/> |
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5798 </head> |
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5799 <pb file="add_6787_f389v" o="389v" n="777"/> |
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5800 <pb file="add_6787_f390" o="390" n="778"/> |
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5801 <pb file="add_6787_f390v" o="390v" n="779"/> |
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5802 <pb file="add_6787_f391" o="391" n="780"/> |
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5803 <pb file="add_6787_f391v" o="391v" n="781"/> |
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5804 <pb file="add_6787_f392" o="392" n="782"/> |
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5805 <pb file="add_6787_f392v" o="392v" n="783"/> |
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5806 <pb file="add_6787_f393" o="393" n="784"/> |
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5807 <pb file="add_6787_f393v" o="393v" n="785"/> |
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5808 <pb file="add_6787_f394" o="394" n="786"/> |
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5809 <div xml:id="echoid-div116" type="page_commentary" level="2" n="116"> |
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5810 <p> |
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5811 <s xml:id="echoid-s532" xml:space="preserve">[<emph style="it">Note: |
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5812 <p> |
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5813 <s xml:id="echoid-s532" xml:space="preserve"> |
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5814 The reference on this page is to Giambattista Benedetti, |
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5815 <emph style="it">Diversarum speculationum mathematicarum et physicarum liber</emph> (1585). |
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5816 </s> |
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5817 </p> |
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5818 </emph>] |
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5819 <lb/><lb/></s></p></div> |
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5820 <head xml:id="echoid-head113" xml:space="preserve" xml:lang="lat"> |
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5821 Locus in sphaeram concavu et convexa, videtur in catheto. <lb/> |
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5822 fallitur Baptista de Ben: <lb/> |
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5823 pag: 339 <lb/> |
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5824 343 <lb/> |
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5825 344 |
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5826 </head> |
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5827 <pb file="add_6787_f394v" o="394v" n="787"/> |
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5828 <pb file="add_6787_f395" o="395" n="788"/> |
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5829 <pb file="add_6787_f395v" o="395v" n="789"/> |
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5830 <div xml:id="echoid-div117" type="page_commentary" level="2" n="117"> |
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5831 <p> |
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5832 <s xml:id="echoid-s534" xml:space="preserve">[<emph style="it">Note: |
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5833 <p> |
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5834 <s xml:id="echoid-s534" xml:space="preserve"> |
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5835 Some rough work on Proposition 14 from Chapter XIX of Viète's |
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5836 <emph style="it">Variorum responsorum liber VIII</emph>, |
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5837 continued from Add MS 6787, f. 227 to f. 230. |
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5838 </s> |
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5839 </p> |
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5840 </emph>] |
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5841 <lb/><lb/></s></p></div> |
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5842 <head xml:id="echoid-head114" xml:space="preserve"> |
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5843 Viet. lib. 8. res. prop. 14. <foreign xml:lang="gre">proch?on</foreign> <lb/> |
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5844 pag. 35. |
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5845 <lb/>[<emph style="it">tr: |
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5846 Viète, Responsorum liber VIII, Proposition 14, page 35. |
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5847 </emph>]<lb/> |
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5848 </head> |
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5849 <pb file="add_6787_f396" o="396" n="790"/> |
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5850 <pb file="add_6787_f396v" o="396v" n="791"/> |
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5851 <pb file="add_6787_f397" o="397" n="792"/> |
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5852 <pb file="add_6787_f397v" o="397v" n="793"/> |
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5853 <pb file="add_6787_f398" o="398" n="794"/> |
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5854 <pb file="add_6787_f398v" o="398v" n="795"/> |
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5855 <div xml:id="echoid-div118" type="page_commentary" level="2" n="118"> |
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5856 <p> |
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5857 <s xml:id="echoid-s536" xml:space="preserve">[<emph style="it">Note: |
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5858 <p> |
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5859 <s xml:id="echoid-s536" xml:space="preserve">The reference on this page is to Viète's |
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5860 <emph style="it">Variorum de rebus mathematicis responsorum lilber VIII</emph> (1593). |
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5861 Viète investigated the properties of the quadratrix in Chapter VII. |
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5862 Harriot's diagram is the same as Viète's. |
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5863 </s> |
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5864 </p> |
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5865 </emph>] |
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5866 <lb/><lb/></s></p></div> |
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5867 <head xml:id="echoid-head115" xml:space="preserve" xml:lang="lat"> |
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5868 Vieta resp. pag. 12. lin: 9, De quadrataria |
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5869 <lb/>[<emph style="it">tr: |
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5870 Viète, Responsorum, page 12, line 9, On the quadratirix |
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5871 </emph>]<lb/> |
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5872 </head> |
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5873 <p xml:lang="lat"> |
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5874 <s xml:id="echoid-s538" xml:space="preserve"> |
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5875 Sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi><mo>,</mo><mi>A</mi><mi>B</mi><mo>:</mo><mi>A</mi><mi>B</mi><mo>,</mo><mi>A</mi><mi>E</mi></mstyle></math>. <lb/> |
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5876 <lb/>[...]<lb/> |
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5877 <lb/>[<emph style="it">tr: |
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5878 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi><mo>:</mo><mi>A</mi><mi>B</mi><mo>=</mo><mi>A</mi><mi>B</mi><mo>:</mo><mi>A</mi><mi>E</mi></mstyle></math>. <lb/> |
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5879 <lb/>[...]<lb/> |
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5880 </emph>]<lb/> |
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5881 </s> |
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5882 </p> |
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5883 <p xml:lang="lat"> |
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5884 <s xml:id="echoid-s539" xml:space="preserve"> |
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5885 Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>E</mi></mstyle></math> maior, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math> <lb/> |
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5886 <lb/>[...]<lb/> <lb/> |
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5887 Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>H</mi><mo>=</mo><mi>F</mi><mi>E</mi></mstyle></math> absurdum |
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5888 <lb/>[<emph style="it">tr: |
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5889 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>E</mi></mstyle></math> be greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math>. <lb/> |
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5890 <lb/>[...]<lb/> <lb/> |
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5891 Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>H</mi><mo>=</mo><mi>F</mi><mi>E</mi></mstyle></math>, absurd |
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5892 </emph>]<lb/> |
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5893 </s> |
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5894 </p> |
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5895 <p xml:lang="lat"> |
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5896 <s xml:id="echoid-s540" xml:space="preserve"> |
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5897 Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>E</mi></mstyle></math> minor, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math> <lb/> |
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5898 <lb/>[...]<lb/> <lb/> |
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5899 Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>E</mi><mo>=</mo><mi>F</mi><mi>E</mi></mstyle></math> absurdum |
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5900 <lb/>[<emph style="it">tr: |
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5901 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>E</mi></mstyle></math> be less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math>. <lb/> |
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5902 <lb/>[...]<lb/> <lb/> |
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5903 Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>E</mi><mo>=</mo><mi>F</mi><mi>E</mi></mstyle></math>, absurd |
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5904 </emph>]<lb/> |
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5905 </s> |
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5906 </p> |
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5907 <p xml:lang="lat"> |
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5908 <s xml:id="echoid-s541" xml:space="preserve"> |
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5909 Et inde: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi><mo>,</mo><mi>A</mi><mi>B</mi><mo>:</mo><mi>A</mi><mi>B</mi><mo>,</mo><mi>B</mi><mi>D</mi></mstyle></math> quæsitum |
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5910 <lb/>[<emph style="it">tr: |
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5911 And hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi><mo>:</mo><mi>A</mi><mi>B</mi><mo>=</mo><mi>A</mi><mi>B</mi><mo>:</mo><mi>B</mi><mi>D</mi></mstyle></math>, which was sought. |
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5912 </emph>]<lb/> |
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5913 </s> |
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5914 </p> |
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5915 <pb file="add_6787_f399" o="399" n="796"/> |
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5916 <pb file="add_6787_f399v" o="399v" n="797"/> |
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5917 <pb file="add_6787_f400" o="400" n="798"/> |
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5918 <pb file="add_6787_f400v" o="400v" n="799"/> |
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5919 <div xml:id="echoid-div119" type="page_commentary" level="2" n="119"> |
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5920 <p> |
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5921 <s xml:id="echoid-s542" xml:space="preserve">[<emph style="it">Note: |
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5922 <p> |
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5923 <s xml:id="echoid-s542" xml:space="preserve"> |
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|
5924 This page refers to Proposition 51 of Book III of Apollonius, as edited by Commandino in |
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|
5925 <emph style="it">Conicorum libri quattuor</emph> (1566). |
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5926 </s> |
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5927 <lb/> |
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|
5928 <quote> |
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|
5929 III.51 |
|
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|
5930 If a rectangle equal to a fourth part of the figure is applied from both sides to the axis of an hyperbola |
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5931 or opposite sections and exceeding by a square figure, and straight lines are deflected from |
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|
5932 the resulting points of application to either one of the sections, |
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5933 then the greater of the two straight lines exceeds the less by exactly as much as the axis. |
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5934 </quote> |
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5935 </p> |
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5936 </emph>] |
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5937 <lb/><lb/></s></p></div> |
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5938 <head xml:id="echoid-head116" xml:space="preserve" xml:lang="lat"> |
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5939 Hyperboles descriptio. per 51. pr. 3: lib. Appollonij |
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5940 <lb/>[<emph style="it">tr: |
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5941 A description of a hyperbola, by Propostion 53, Book 3, of Apollonius. |
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5942 </emph>]<lb/> |
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5943 </head> |
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5944 <pb file="add_6787_f401" o="401" n="800"/> |
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5945 <pb file="add_6787_f401v" o="401v" n="801"/> |
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5946 <pb file="add_6787_f402" o="402" n="802"/> |
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5947 <pb file="add_6787_f402v" o="402v" n="803"/> |
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5948 <div xml:id="echoid-div120" type="page_commentary" level="2" n="120"> |
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5949 <p> |
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5950 <s xml:id="echoid-s544" xml:space="preserve">[<emph style="it">Note: |
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5951 <p> |
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5952 <s xml:id="echoid-s544" xml:space="preserve"> |
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5953 The reference on this page is to Giambattista Benedetti, |
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5954 <emph style="it">Diversarum speculationum mathematicarum et physicarum liber</emph> (1585). |
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5955 </s> |
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5956 </p> |
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5957 </emph>] |
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5958 <lb/><lb/></s></p></div> |
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5959 <head xml:id="echoid-head117" xml:space="preserve" xml:lang="lat"> |
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5960 Bap: de Ben: <lb/> |
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5961 pag: 349 <lb/> |
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5962 334 |
|
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5963 <lb/>[<emph style="it">tr: |
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5964 Baptista de Benedictis, page 349, 334 |
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5965 </emph>]<lb/> |
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5966 </head> |
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5967 <pb file="add_6787_f403" o="403" n="804"/> |
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5968 <pb file="add_6787_f403v" o="403v" n="805"/> |
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5969 <pb file="add_6787_f404" o="404" n="806"/> |
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5970 <div xml:id="echoid-div121" type="page_commentary" level="2" n="121"> |
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5971 <p> |
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5972 <s xml:id="echoid-s546" xml:space="preserve">[<emph style="it">Note: |
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5973 <p> |
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5974 <s xml:id="echoid-s546" xml:space="preserve"> |
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5975 The references on this page are to Francesco Barozzi, |
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5976 <emph style="it">Admirandum illud geometricum problema tredecim modis demonstratum </emph> (1586). |
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|
5977 Page 104 contains a diagram. |
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5978 Harriot's (lower case) letters correspond to the (upper case) letters in Barozzi's diagram. |
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5979 </s> |
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5980 </p> |
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5981 </emph>] |
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5982 <lb/><lb/></s></p></div> |
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5983 <head xml:id="echoid-head118" xml:space="preserve" xml:lang="lat"> |
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|
5984 Barocius. pag. 104. |
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5985 </head> |
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5986 <pb file="add_6787_f404v" o="404v" n="807"/> |
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5987 <pb file="add_6787_f405" o="405" n="808"/> |
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5988 <pb file="add_6787_f405v" o="405v" n="809"/> |
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5989 <pb file="add_6787_f406" o="406" n="810"/> |
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5990 <pb file="add_6787_f406v" o="406v" n="811"/> |
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5991 <pb file="add_6787_f407" o="407" n="812"/> |
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5992 <pb file="add_6787_f407v" o="407v" n="813"/> |
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5993 <pb file="add_6787_f408" o="408" n="814"/> |
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5994 <pb file="add_6787_f408v" o="408v" n="815"/> |
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5995 <pb file="add_6787_f409" o="409" n="816"/> |
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5996 <pb file="add_6787_f409v" o="409v" n="817"/> |
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5997 <pb file="add_6787_f410" o="410" n="818"/> |
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5998 <pb file="add_6787_f410v" o="410v" n="819"/> |
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5999 <pb file="add_6787_f411" o="411" n="820"/> |
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6000 <pb file="add_6787_f411v" o="411v" n="821"/> |
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6001 <pb file="add_6787_f412" o="412" n="822"/> |
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6002 <pb file="add_6787_f412v" o="412v" n="823"/> |
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6003 <pb file="add_6787_f413" o="413" n="824"/> |
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6004 <pb file="add_6787_f413v" o="413v" n="825"/> |
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6005 <pb file="add_6787_f414" o="414" n="826"/> |
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6006 <pb file="add_6787_f414v" o="414v" n="827"/> |
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6007 <pb file="add_6787_f415" o="415" n="828"/> |
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6008 <pb file="add_6787_f415v" o="415v" n="829"/> |
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6009 <pb file="add_6787_f416" o="416" n="830"/> |
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6010 <pb file="add_6787_f416v" o="416v" n="831"/> |
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6011 <pb file="add_6787_f417" o="417" n="832"/> |
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6012 <pb file="add_6787_f417v" o="417v" n="833"/> |
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6013 <pb file="add_6787_f418" o="418" n="834"/> |
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6014 <pb file="add_6787_f418v" o="418v" n="835"/> |
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6015 <pb file="add_6787_f419" o="419" n="836"/> |
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6016 <pb file="add_6787_f419v" o="419v" n="837"/> |
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6017 <pb file="add_6787_f420" o="420" n="838"/> |
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6018 <pb file="add_6787_f420v" o="420v" n="839"/> |
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6019 <pb file="add_6787_f421" o="421" n="840"/> |
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6020 <pb file="add_6787_f421v" o="421v" n="841"/> |
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6021 <pb file="add_6787_f422" o="422" n="842"/> |
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6022 <pb file="add_6787_f422v" o="422v" n="843"/> |
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6023 <pb file="add_6787_f423" o="423" n="844"/> |
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6024 <pb file="add_6787_f423v" o="423v" n="845"/> |
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6025 <pb file="add_6787_f424" o="424" n="846"/> |
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6026 <pb file="add_6787_f424v" o="424v" n="847"/> |
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6027 <pb file="add_6787_f425" o="425" n="848"/> |
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6028 <div xml:id="echoid-div122" type="page_commentary" level="2" n="122"> |
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6029 <p> |
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6030 <s xml:id="echoid-s548" xml:space="preserve">[<emph style="it">Note: |
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6031 <p> |
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6032 <s xml:id="echoid-s548" xml:space="preserve"> |
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6033 The reference towards the end of this page is to Giambattista Benedetti, |
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6034 <emph style="it">Diversarum speculationum mathematicarum et physicarum liber</emph> (1585), |
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6035 thought the theorem on page 26 is actually Theorem 41, not Theorem 45. <lb/> |
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6036 There is also a reference to Proposition 13 from Viète's |
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6037 <emph style="it">Effectionum geometricarum canonica recensio</emph> (1593), |
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6038 which explains how to find two quantities from their geometric mean and their sum. |
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6039 </s> |
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6040 </p> |
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6041 </emph>] |
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6042 <lb/><lb/></s></p></div> |
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6043 <p xml:lang="lat"> |
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6044 <s xml:id="echoid-s550" xml:space="preserve"> |
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6045 Data |
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6046 <lb/>[<emph style="it">tr: |
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6047 Given |
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6048 </emph>]<lb/> |
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6049 </s> |
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6050 </p> |
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6051 <p xml:lang="lat"> |
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6052 <s xml:id="echoid-s551" xml:space="preserve"> |
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6053 Data in partibus canonis |
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6054 <lb/>[<emph style="it">tr: |
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6055 Given in parts of the canon |
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6056 </emph>]<lb/> |
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6057 </s> |
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6058 </p> |
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6059 <p xml:lang="lat"> |
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6060 <s xml:id="echoid-s552" xml:space="preserve"> |
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6061 Ex Methodo <lb/> |
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6062 Adde [???] <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>a</mi><mi>d</mi></mstyle></math> <lb/> |
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6063 vel illa æqualium <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>b</mi><mi>o</mi></mstyle></math> <lb/> |
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6064 qui in centro <lb/> |
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6065 sed in principia |
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6066 <lb/>[<emph style="it">tr: |
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6067 By the method: add <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>a</mi><mi>d</mi></mstyle></math> or its equal <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>b</mi><mi>o</mi></mstyle></math> which in the centre is <lb/>[...]<lb/> but in the beginning <lb/>[...]<lb/>. |
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6068 </emph>]<lb/> |
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6069 <lb/>[<emph style="it">tr: |
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6070 Given |
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6071 </emph>]<lb/> |
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6072 </s> |
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6073 </p> |
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6074 <p xml:lang="lat"> |
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6075 <s xml:id="echoid-s553" xml:space="preserve"> |
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6076 tum quæritur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>n</mi></mstyle></math> et inde <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>l</mi></mstyle></math>. per Theor. 45. pag. 26. Joh. Baptistæ <lb/> |
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|
6077 de Benedictis |
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6078 <lb/>[<emph style="it">tr: |
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6079 then there is sought <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>n</mi></mstyle></math> and hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>l</mi></mstyle></math>, by Theorem 45, page 26, Johan Baptista de Benedictis |
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6080 </emph>]<lb/> |
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6081 </s> |
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6082 </p> |
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6083 <p xml:lang="lat"> |
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6084 <s xml:id="echoid-s554" xml:space="preserve"> |
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6085 Vel per Algebræ |
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6086 <lb/>[<emph style="it">tr: |
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6087 Or by algebra |
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6088 </emph>]<lb/> |
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6089 </s> |
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6090 <lb/> |
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6091 <s xml:id="echoid-s555" xml:space="preserve"> |
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6092 Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>n</mi></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>r</mi></mstyle></math>. tum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>b</mi></mstyle></math> erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>-</mo><mn>1</mn><mi>r</mi></mstyle></math>. hoc multiplicatum per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>r</mi></mstyle></math> faciet <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mi>r</mi><mo>-</mo><mn>1</mn><mi>q</mi></mstyle></math>. <lb/> |
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6093 quod æquale erit rectangulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>n</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>m</mi></mstyle></math>, hoc est 78,545,532. <lb/> |
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6094 Forma æquationis ita erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mi>r</mi><mo>-</mo><mn>1</mn><mi>q</mi><mo>=</mo><mn>7</mn><mn>8</mn><mo>,</mo><mn>5</mn><mn>4</mn><mn>5</mn><mo>,</mo><mn>5</mn><mn>3</mn><mn>2</mn></mstyle></math>. <lb/> |
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6095 Et duplis erit responsum, videlicet |
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6096 <lb/>[<emph style="it">tr: |
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6097 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>n</mi><mo>=</mo><mn>1</mn><mi>r</mi></mstyle></math>. then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>b</mi></mstyle></math> will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mo>-</mo><mn>1</mn><mi>r</mi></mstyle></math>. This multiplied by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>r</mi></mstyle></math> makes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mi>r</mi><mo>-</mo><mn>1</mn><mi>q</mi></mstyle></math>. <lb/> |
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6098 which is equal to the rectangle of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>n</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>m</mi></mstyle></math>, that is 78,545,532. <lb/> |
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6099 Thus the form of the equation will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>0</mn><mo>,</mo><mn>0</mn><mn>0</mn><mn>0</mn><mi>r</mi><mo>-</mo><mn>1</mn><mi>q</mi><mo>=</mo><mn>7</mn><mn>8</mn><mo>,</mo><mn>5</mn><mn>4</mn><mn>5</mn><mo>,</mo><mn>5</mn><mn>3</mn><mn>2</mn></mstyle></math>. <lb/> |
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6100 And it will be twice the answer, namely |
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6101 </emph>]<lb/> |
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6102 </s> |
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6103 <lb/> |
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6104 <s xml:id="echoid-s556" xml:space="preserve"> |
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6105 Habetur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>n</mi></mstyle></math> alias per 13 prop. Geom. Effect. <lb/> |
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6106 Vietæ |
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6107 <lb/>[<emph style="it">tr: |
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6108 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>n</mi></mstyle></math> can be had otherwise by Proposition 13 of Vi´te, <emph style="it">Effectionum geometricarum</emph>. |
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6109 </emph>]<lb/> |
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6110 </s> |
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6111 </p> |
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6112 <pb file="add_6787_f425v" o="425v" n="849"/> |
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6113 <pb file="add_6787_f426" o="426" n="850"/> |
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6114 <pb file="add_6787_f426v" o="426v" n="851"/> |
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6115 <pb file="add_6787_f427" o="427" n="852"/> |
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6116 <pb file="add_6787_f427v" o="427v" n="853"/> |
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6117 <pb file="add_6787_f428" o="428" n="854"/> |
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6118 <pb file="add_6787_f428v" o="428v" n="855"/> |
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6119 <pb file="add_6787_f429" o="429" n="856"/> |
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6120 <pb file="add_6787_f429v" o="429v" n="857"/> |
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6121 <pb file="add_6787_f430" o="430" n="858"/> |
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6122 <pb file="add_6787_f430v" o="430v" n="859"/> |
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6123 <pb file="add_6787_f431" o="431" n="860"/> |
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6124 <pb file="add_6787_f431v" o="431v" n="861"/> |
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6125 <pb file="add_6787_f432" o="432" n="862"/> |
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6126 <pb file="add_6787_f432v" o="432v" n="863"/> |
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6127 <pb file="add_6787_f433" o="433" n="864"/> |
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6128 <pb file="add_6787_f433v" o="433v" n="865"/> |
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6129 <pb file="add_6787_f434" o="434" n="866"/> |
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6130 <pb file="add_6787_f434v" o="434v" n="867"/> |
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6131 <pb file="add_6787_f435" o="435" n="868"/> |
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6132 <pb file="add_6787_f435v" o="435v" n="869"/> |
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6133 <pb file="add_6787_f436" o="436" n="870"/> |
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6134 <pb file="add_6787_f436v" o="436v" n="871"/> |
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6135 <pb file="add_6787_f437" o="437" n="872"/> |
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6136 <pb file="add_6787_f437v" o="437v" n="873"/> |
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6137 <pb file="add_6787_f438" o="438" n="874"/> |
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6138 <pb file="add_6787_f438v" o="438v" n="875"/> |
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6139 <pb file="add_6787_f439" o="439" n="876"/> |
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6140 <pb file="add_6787_f439v" o="439v" n="877"/> |
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6141 <pb file="add_6787_f440" o="440" n="878"/> |
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6142 <pb file="add_6787_f440v" o="440v" n="879"/> |
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6143 <pb file="add_6787_f441" o="441" n="880"/> |
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6144 <pb file="add_6787_f441v" o="441v" n="881"/> |
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6145 <pb file="add_6787_f442" o="442" n="882"/> |
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6146 <pb file="add_6787_f442v" o="442v" n="883"/> |
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6147 <pb file="add_6787_f443" o="443" n="884"/> |
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6148 <pb file="add_6787_f443v" o="443v" n="885"/> |
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6149 <pb file="add_6787_f444" o="444" n="886"/> |
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6150 <pb file="add_6787_f444v" o="444v" n="887"/> |
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6151 <pb file="add_6787_f445" o="445" n="888"/> |
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6152 <pb file="add_6787_f445v" o="445v" n="889"/> |
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6153 <pb file="add_6787_f446" o="446" n="890"/> |
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6154 <pb file="add_6787_f446v" o="446v" n="891"/> |
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6155 <pb file="add_6787_f447" o="447" n="892"/> |
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6156 <pb file="add_6787_f447v" o="447v" n="893"/> |
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6157 <pb file="add_6787_f448" o="448" n="894"/> |
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6158 <pb file="add_6787_f448v" o="448v" n="895"/> |
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6159 <pb file="add_6787_f449" o="449" n="896"/> |
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6160 <pb file="add_6787_f449v" o="449v" n="897"/> |
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6161 <div xml:id="echoid-div123" type="page_commentary" level="2" n="123"> |
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6162 <p> |
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6163 <s xml:id="echoid-s557" xml:space="preserve">[<emph style="it">Note: |
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6164 <p> |
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6165 <s xml:id="echoid-s557" xml:space="preserve"> |
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6166 This page refers to Proposition 37 of Book III of Apollonius, as edited by Commandino in |
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6167 <emph style="it">Conicorum libri quattuor</emph> (1566). |
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6168 </s> |
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6169 <lb/> |
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6170 <quote> |
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6171 If two straight lines touching a section of a cone or circumference of a circle or opposite sections meet, |
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6172 and a straight line is joined to their points of contact, and from the point of meeting of the tangents |
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6173 some straight lin is drawn across cutting the line (of the section) at two points, |
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6174 then as the whole straight line is to the straight line cut off from outside, |
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6175 so will the segments proeuced by the straight line joining the points be to each other. |
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6176 </quote> |
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6177 </p> |
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6178 </emph>] |
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6179 <lb/><lb/></s></p></div> |
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6180 <p xml:lang="lat"> |
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6181 <s xml:id="echoid-s559" xml:space="preserve"> |
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6182 37.3<emph style="super">i</emph> |
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6183 <lb/>[<emph style="it">tr: |
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6184 Proposition III.37 |
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6185 </emph>]<lb/> |
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6186 </s> |
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6187 </p> |
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6188 <pb file="add_6787_f450" o="450" n="898"/> |
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6189 <pb file="add_6787_f450v" o="450v" n="899"/> |
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6190 <pb file="add_6787_f451" o="451" n="900"/> |
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6191 <div xml:id="echoid-div124" type="page_commentary" level="2" n="124"> |
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6192 <p> |
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6193 <s xml:id="echoid-s560" xml:space="preserve">[<emph style="it">Note: |
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6194 <p> |
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6195 <s xml:id="echoid-s560" xml:space="preserve"> |
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6196 This page refers to Propositions I.21 and I.34 of Apollonius, as edited by Commandino in |
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6197 <emph style="it">Conicorum libri quattuor</emph> (1566). |
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6198 </s> |
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6199 <lb/> |
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6200 <quote> |
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6201 I.21 |
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6202 If in a hyperbola or ellipse or circumference of a circle straight lines are dropped as ordinates to the diameter, |
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6203 the square on them will be to the areas contained by the straight lines cut off by them beginning from the ends of |
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6204 the transverse side of the figure, as the upright side of the figure is to the transverse, |
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6205 and to each other as the areas contained by the straight lines cut off, as we have said. |
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6206 </quote> |
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6207 <lb/> |
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6208 <quote> |
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6209 I.34 If on a hyperbola or ellipse or circumference of a circle some point is taken, |
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6210 and from it a straight line is dropped as ordinate to the diameter, |
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6211 and if the straight lines which the ordinate cuts off from the ends of the figure 19s transverse side |
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6212 have to each other a ratio which other segments of the transverse side have to each other, |
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6213 so that the segments from the vertex are corresponding, then the straight line joining the point |
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6214 taken on the transverse side and that taken on the section will touch the section. |
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6215 </quote> |
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6216 </p> |
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6217 </emph>] |
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6218 <lb/><lb/></s></p></div> |
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6219 <head xml:id="echoid-head119" xml:space="preserve" xml:lang="lat"> |
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6220 Appol. lib: 1. pag. 25 <lb/> |
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6221 prop: 34. Elementa tactus |
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6222 <lb/>[<emph style="it">tr: |
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6223 Apollonius, Book 1, page 25, Proposition 34. Elements of tangents. |
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6224 </emph>]<lb/> |
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6225 </head> |
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6226 <p xml:lang="lat"> |
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6227 <s xml:id="echoid-s562" xml:space="preserve"> |
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6228 Sit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>a</mi></mstyle></math> : <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi></mstyle></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>a</mi></mstyle></math> <lb/> |
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6229 iugatur: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>c</mi></mstyle></math> <lb/> |
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6230 Dico quod <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>c</mi></mstyle></math> tangit <lb/> |
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6231 si non secet ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>c</mi><mi>f</mi></mstyle></math> |
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6232 <lb/>[<emph style="it">tr: |
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6233 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</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>b</mi><mi>e</mi></mstyle></math> : <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>a</mi></mstyle></math>, and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>c</mi></mstyle></math> be joined; I say that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>c</mi></mstyle></math> is a tangent. <lb/> |
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6234 If not, it cuts, as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>c</mi><mi>f</mi></mstyle></math>. |
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6235 </emph>]<lb/> |
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6236 </s> |
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6237 </p> |
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6238 <p xml:lang="lat"> |
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6239 <s xml:id="echoid-s563" xml:space="preserve"> |
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6240 per 21, 1 <lb/>[...]<lb/> ex hypoth |
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6241 <lb/>[<emph style="it">tr: |
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6242 by Propositon I.21 <lb/>[...]<lb/> by hypothesis |
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6243 </emph>]<lb/> |
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6244 </s> |
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6245 </p> |
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6246 <p xml:lang="lat"> |
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6247 <s xml:id="echoid-s564" xml:space="preserve"> |
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6248 absurdum |
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6249 <lb/>[<emph style="it">tr: |
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6250 absurd |
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6251 </emph>]<lb/> |
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6252 </s> |
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6253 </p> |
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6254 <pb file="add_6787_f451v" o="451v" n="901"/> |
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6255 <pb file="add_6787_f452" o="452" n="902"/> |
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6256 <pb file="add_6787_f452v" o="452v" n="903"/> |
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6257 <pb file="add_6787_f453" o="453" n="904"/> |
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6258 <pb file="add_6787_f453v" o="453v" n="905"/> |
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6259 <pb file="add_6787_f454" o="454" n="906"/> |
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6260 <pb file="add_6787_f454v" o="454v" n="907"/> |
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6261 <pb file="add_6787_f455" o="455" n="908"/> |
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6262 <div xml:id="echoid-div125" type="page_commentary" level="2" n="125"> |
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6263 <p> |
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6264 <s xml:id="echoid-s565" xml:space="preserve">[<emph style="it">Note: |
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6265 <p> |
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6266 <s xml:id="echoid-s565" xml:space="preserve"> |
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6267 This page refers to Proposition 17 of Book III of Apollonius, as edited by Commandino in |
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6268 <emph style="it">Conicorum libri quattuor</emph> (1566). |
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6269 </s> |
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6270 <lb/> |
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6271 <quote> |
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6272 III. 17 |
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6273 If two straight lines touching a section of a cone or circumference of a circle meet, |
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6274 and two points are taken at random on the section, and from them in the section are drawn parallel |
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6275 to the tangents straight lines cutting each other and the line of the section, |
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6276 then as the squares on the tangents are to each other, so will the rectangles contained by the straight lines |
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6277 taken similarly. |
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6278 </quote> |
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6279 </p> |
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6280 </emph>] |
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6281 <lb/><lb/></s></p></div> |
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6282 <head xml:id="echoid-head120" xml:space="preserve" xml:lang="lat"> |
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6283 1) circa 5 data puncta <emph style="super">quæ sunt in ellipsi</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>k</mi><mi>l</mi><mi>m</mi><mi>n</mi></mstyle></math> <lb/> |
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6284 ellipsin describere. |
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6285 <lb/>[<emph style="it">tr: |
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6286 Around 5 given points which are on an ellipse <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>k</mi><mi>l</mi><mi>m</mi><mi>n</mi></mstyle></math>, describe the ellipse. |
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6287 </emph>]<lb/> |
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6288 </head> |
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6289 <p xml:lang="lat"> |
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6290 <s xml:id="echoid-s567" xml:space="preserve"> |
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6291 sit descripta. |
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6292 <lb/>[<emph style="it">tr: |
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6293 suppose it described |
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6294 </emph>]<lb/> |
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6295 </s> |
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6296 <lb/> |
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6297 <s xml:id="echoid-s568" xml:space="preserve"> |
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6298 1. casus. sint <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>k</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>h</mi></mstyle></math>, parallelæ. <lb/> |
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6299 dividantur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>k</mi></mstyle></math>, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>h</mi></mstyle></math> bisarium <lb/> |
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6300 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>a</mi></mstyle></math>. et per <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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6301 ducatur recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>e</mi></mstyle></math>. quæ diameter <lb/> |
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6302 est ellipseos <lb/> |
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6303 data <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>m</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>h</mi></mstyle></math> |
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6304 <lb/>[<emph style="it">tr: |
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6305 Case 1. Suppose <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>k</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>h</mi></mstyle></math> are parallel. <lb/> |
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6306 Bisect <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>k</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>h</mi></mstyle></math> at the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, and through <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> is drawn the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>e</mi></mstyle></math>, |
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6307 which is the diameter of the ellipse. <lb/> |
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6308 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>m</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>h</mi></mstyle></math> are given. |
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6309 </emph>]<lb/> |
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6310 </s> |
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6311 </p> |
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6312 <p xml:lang="lat"> |
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6313 <s xml:id="echoid-s569" xml:space="preserve"> |
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6314 17.3.coni |
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6315 <lb/>[<emph style="it">tr: |
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6316 Proposition III.17 of the conics. |
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6317 </emph>]<lb/> |
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6318 </s> |
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6319 </p> |
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6320 <p xml:lang="lat"> |
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6321 <s xml:id="echoid-s570" xml:space="preserve"> |
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6322 datur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>c</mi><mo>.</mo><mi>c</mi><mi>g</mi></mstyle></math> quoniam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi><mi>c</mi><mi>x</mi></mstyle></math> parallela <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>c</mi></mstyle></math>, secat <lb/> |
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6323 datam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>h</mi></mstyle></math> in puncto <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>c</mi></mstyle></math> ergo datam et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>c</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>h</mi></mstyle></math>. <lb/> |
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6324 ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> datam quoniam |
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6325 <lb/>[<emph style="it">tr: |
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6326 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>c</mi><mo>.</mo><mi>c</mi><mi>g</mi></mstyle></math> is given because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi><mi>c</mi><mi>x</mi></mstyle></math> is parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>c</mi></mstyle></math>, and cuts the given line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>h</mi></mstyle></math> in the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. |
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6327 The point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> is therefore given, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>c</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>h</mi></mstyle></math>. <lb/> |
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6328 Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> is geven because |
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6329 ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> datam quoniam </emph>]<lb/> |
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6330 </s> |
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6331 </p> |
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6332 <p xml:lang="lat"> |
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6333 <s xml:id="echoid-s571" xml:space="preserve"> |
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6334 sed et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math> datur. Igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>p</mi><mi>x</mi></mstyle></math> postione <lb/> |
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6335 et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi><mi>c</mi><mi>x</mi></mstyle></math> positione. Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math>, dabitur, quod est in ellipsi. |
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6336 <lb/>[<emph style="it">tr: |
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6337 but also <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math> is given. Therefore the position of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>p</mi><mi>x</mi></mstyle></math> and the position of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi><mi>c</mi><mi>x</mi></mstyle></math>. |
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6338 Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi></mstyle></math> will be given, which is on the ellipse. |
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6339 </emph>]<lb/> |
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6340 </s> |
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6341 </p> |
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6342 <p xml:lang="lat"> |
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6343 <s xml:id="echoid-s572" xml:space="preserve"> |
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6344 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>a</mi><mo>.</mo><mi>a</mi><mi>s</mi></mstyle></math> datum. quia, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>r</mi></mstyle></math>, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>s</mi></mstyle></math>, data. <lb/> |
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6345 Igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi><mo>.</mo><mi>e</mi><mi>f</mi></mstyle></math> datur. |
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6346 <lb/>[<emph style="it">tr: |
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6347 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>a</mi><mo>.</mo><mi>a</mi><mi>s</mi></mstyle></math> is fiven, because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>r</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>s</mi></mstyle></math> are given. <lb/> |
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6348 Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi><mo>.</mo><mi>e</mi><mi>f</mi></mstyle></math> is given. </emph>]<lb/> |
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6349 </s> |
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6350 </p> |
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6351 <pb file="add_6787_f455v" o="455v" n="909"/> |
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6352 <pb file="add_6787_f456" o="456" n="910"/> |
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6353 <pb file="add_6787_f456v" o="456v" n="911"/> |
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6354 <pb file="add_6787_f457" o="457" n="912"/> |
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6355 <pb file="add_6787_f457v" o="457v" n="913"/> |
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6356 <pb file="add_6787_f458" o="458" n="914"/> |
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6357 <div xml:id="echoid-div126" type="page_commentary" level="2" n="126"> |
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6358 <p> |
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6359 <s xml:id="echoid-s573" xml:space="preserve">[<emph style="it">Note: |
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6360 <p> |
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6361 <s xml:id="echoid-s573" xml:space="preserve"> |
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6362 This page refers to Propositions I.21 and I.54 of Apollonius, as edited by Commandino in |
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6363 <emph style="it">Conicorum libri quattuor</emph> (1566). |
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6364 </s> |
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6365 <lb/> |
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6366 <quote> |
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6367 I.21 |
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6368 If in a hyperbola or ellipse or circumference of a circle straight lines are dropped as ordinates to the diameter, |
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6369 the square on them will be to the areas contained by the straight lines cut off by them beginning from the ends of |
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6370 the transverse side of the figure, as the upright side of the figure is to the transverse, |
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6371 and to each other as the areas contained by the straight lines cut off, as we have said. |
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6372 </quote> |
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6373 <lb/> |
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6374 <quote> |
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6375 I.54 Given two bounded straight lines perpendicular to each other, |
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6376 one of them being produced on the side of the right angle, |
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6377 to find on the straight line produced the section of a cone called hyperbola in the same plane with the straight line, |
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6378 so that the straight line produced is a diameter of the section and the point at the angle is the vertex, |
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6379 and where whatever straight line is dropped from the section to the diameter, making an angle equal to a given angle, |
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6380 will equal in square the rectangle applied to the other straight line having as breadth |
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6381 the straight line cut off the dropped straight line beginning with the vertex and projecting beyond |
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6382 by a figure similar and similarly situated to that contained by the original straight lines. |
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6383 </quote> |
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6384 </p> |
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6385 </emph>] |
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6386 <lb/><lb/></s></p></div> |
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6387 <head xml:id="echoid-head121" xml:space="preserve" xml:lang="lat"> |
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6388 2) Circa 5 data puncta ellispin describere |
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6389 <lb/>[<emph style="it">tr: |
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6390 To describe an ellipse around five given points. |
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6391 </emph>]<lb/> |
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6392 </head> |
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6393 <p xml:lang="lat"> |
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6394 <s xml:id="echoid-s575" xml:space="preserve"> |
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6395 simili ratione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> dabitur |
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6396 <lb/>[<emph style="it">tr: |
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6397 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> are given in similar ratio |
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6398 </emph>]<lb/> |
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6399 </s> |
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6400 </p> |
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6401 <p xml:lang="lat"> |
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6402 <s xml:id="echoid-s576" xml:space="preserve"> |
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6403 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi><mo>.</mo><mi>b</mi><mi>z</mi></mstyle></math> datur. ergo, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>b</mi><mo>.</mo><mi>b</mi><mi>f</mi></mstyle></math> datur. |
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6404 <lb/>[<emph style="it">tr: |
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6405 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi><mo>.</mo><mi>b</mi><mi>z</mi></mstyle></math> is given, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>b</mi><mo>.</mo><mi>b</mi><mi>f</mi></mstyle></math> is given. |
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6406 </emph>]<lb/> |
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6407 </s> |
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6408 </p> |
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6409 <p xml:lang="lat"> |
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6410 <s xml:id="echoid-s577" xml:space="preserve"> |
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6411 dantur <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>, puncta; ergo, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math> ut deinceps. (alia charta) <lb/> |
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6412 quære <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math> diameter maagnitudine data est, et diameter <lb/> |
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6413 ipsi coniugata, cum datur proportio transversis lateris et ad <lb/> |
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6414 rectum. eadem enim est ut |
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6415 <lb/>[<emph style="it">tr: |
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6416 there are given the points <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>; and thereafter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math> (another sheet); |
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6417 whereby the magnitude of the diameter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math> is given, and the diameter of the conjugate, |
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6418 since the ratio to the transverse side is given and to the latus rectum. For it is the same as: |
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6419 </emph>]<lb/> |
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6420 </s> |
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6421 </p> |
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6422 <p xml:lang="lat"> |
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6423 <s xml:id="echoid-s578" xml:space="preserve"> |
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6424 nam <lb/> |
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6425 21.1.con.) |
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6426 <lb/>[<emph style="it">tr: |
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6427 for by proposition I.21 of the conics |
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6428 </emph>]<lb/> |
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6429 </s> |
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6430 <lb/> |
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6431 <s xml:id="echoid-s579" xml:space="preserve"> |
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6432 ergo data coniugata diameter, hoc est 2<emph style="super">a</emph> diameter nam <lb/> |
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6433 <lb/>[...]<lb/> <lb/> |
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6434 ergo per <lb/> |
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6435 54.1.con <lb/> |
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6436 ellipsis describatur, vel datur |
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6437 <lb/>[<emph style="it">tr: |
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6438 therefore the conjugate diameter is given, that is, the 2nd diameter, for <lb/> |
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6439 <lb/>[...]<lb/> <lb/> |
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6440 therefore by proposition I.54 of the conics, an ellipse is described, or given. |
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6441 </emph>]<lb/> |
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6442 </s> |
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6443 </p> |
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6444 <p xml:lang="lat"> |
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6445 <s xml:id="echoid-s580" xml:space="preserve"> |
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6446 quæritur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> <lb/> |
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6447 et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math> |
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6448 <lb/>[<emph style="it">tr: |
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6449 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> are sought, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math> |
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6450 </emph>]<lb/> |
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6451 </s> |
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6452 </p> |
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6453 <p xml:lang="lat"> |
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6454 <s xml:id="echoid-s581" xml:space="preserve"> |
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6455 [<emph style="it">Note: |
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6456 Sheet 5 is Add MS 6787, f. 461. |
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6457 </emph>]<lb/> |
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6458 in charta <lb/> |
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6459 5 <lb/> |
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6460 ita est |
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6461 <lb/>[<emph style="it">tr: |
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6462 in sheet 5, it is thus |
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6463 </emph>]<lb/> |
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6464 </s> |
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6465 </p> |
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6466 <pb file="add_6787_f458v" o="458v" n="915"/> |
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6467 <pb file="add_6787_f459" o="459" n="916"/> |
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6468 <pb file="add_6787_f459v" o="459v" n="917"/> |
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6469 <pb file="add_6787_f460" o="460" n="918"/> |
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6470 <div xml:id="echoid-div127" type="page_commentary" level="2" n="127"> |
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6471 <p> |
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6472 <s xml:id="echoid-s582" xml:space="preserve">[<emph style="it">Note: |
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6473 <p> |
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6474 <s xml:id="echoid-s582" xml:space="preserve"> |
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6475 The reference to Pappus is to Commandino's edition of Books III to VIII, |
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6476 <emph style="it">Mathematicae collecitones</emph> (1558). |
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6477 The diagram given by Harriot at the top of this folio appears on page 320, |
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6478 as part of Commandino's long commentary to Proposition 13. |
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6479 </s> |
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6480 <lb/> |
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6481 <quote xml:lang="lat"> |
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6482 Problema IX. Propositio XIII. <lb/> |
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6483 Cum autem quæsitum fit circa quinque data puncta HKLMN ellispin describere. |
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6484 Sit iam descripta: & iunctæ MK NH primum sint parallelæ diuidanturque bisariam in punctis AB. |
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6485 & ducta AB ad EF puncta ellipsis producatur. est igitur EF ipsius diameter |
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6486 per diffinitionem conicorum positione data. etenim vnumquodque punctorum AB datum est positione. |
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6487 </quote> |
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6488 <lb/> |
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6489 <quote> |
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6490 Moreover, when it is sought to draw an ellipse about the five given points H, K, L, M, N, suppose it is already done. |
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6491 And first join the lines MK and NH, letting them be parallel, and bisected at the points A and B. |
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6492 And AB is drawn and extended to E and F, points on the ellipse. Therefore, EF is the diameter of that ellipse, |
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6493 by the rules of conics in a given position, and indeed, each of the points A and B is given in position. |
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6494 </quote> |
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6495 </p> |
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6496 </emph>] |
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6497 <lb/><lb/></s></p></div> |
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6498 <head xml:id="echoid-head122" xml:space="preserve"> |
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6499 4.) Circa 5 data puncta <lb/> |
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6500 ellipsin describere. <lb/> |
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6501 pappus. 320 |
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6502 <lb/>[<emph style="it">tr: |
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6503 To draw an ellipse through 5 given points. Pappus, page 320. |
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6504 </emph>]<lb/> |
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6505 </head> |
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6506 <p> |
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6507 <s xml:id="echoid-s584" xml:space="preserve"> |
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6508 melius in <lb/> |
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6509 5 charta |
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6510 <lb/>[<emph style="it">tr: |
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6511 better in sheet 5 |
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6512 </emph>]<lb/> |
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6513 </s> |
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6514 </p> |
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6515 <pb file="add_6787_f460v" o="460v" n="919"/> |
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6516 <pb file="add_6787_f461" o="461" n="920"/> |
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6517 <pb file="add_6787_f461v" o="461v" n="921"/> |
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6518 <pb file="add_6787_f462" o="462" n="922"/> |
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6519 <pb file="add_6787_f462v" o="462v" n="923"/> |
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6520 <pb file="add_6787_f463" o="463" n="924"/> |
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6521 <pb file="add_6787_f463v" o="463v" n="925"/> |
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6522 <pb file="add_6787_f464" o="464" n="926"/> |
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6523 <pb file="add_6787_f464v" o="464v" n="927"/> |
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6524 <pb file="add_6787_f465" o="465" n="928"/> |
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6525 <pb file="add_6787_f465v" o="465v" n="929"/> |
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6526 <pb file="add_6787_f466" o="466" n="930"/> |
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6527 <pb file="add_6787_f466v" o="466v" n="931"/> |
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6528 <pb file="add_6787_f467" o="467" n="932"/> |
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6529 <div xml:id="echoid-div128" type="page_commentary" level="2" n="128"> |
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6530 <p> |
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6531 <s xml:id="echoid-s585" xml:space="preserve">[<emph style="it">Note: |
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6532 <p> |
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6533 <s xml:id="echoid-s585" xml:space="preserve"> |
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6534 The reference on this page is to Guidobaldi del Monte (Guido Ubaldi), |
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6535 <emph style="it">In duos Archimedies aequiponderantium libros paraphrasis</emph> (1588), page 172. |
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6536 </s> |
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6537 </p> |
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6538 </emph>] |
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6539 <lb/><lb/></s></p></div> |
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6540 <head xml:id="echoid-head123" xml:space="preserve" xml:lang="lat"> |
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6541 Archimedes <emph style="st">de quad</emph>: <emph style="super">centro gravitatis</emph> |
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6542 parabolæ Ubaldus. pag: 172. <lb/> |
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6543 de centro gravitatis |
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6544 <lb/>[<emph style="it">tr: |
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6545 Archimedes, on the centre of gravity of a parabola, Ubaldus page 172 |
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6546 </emph>]<lb/> |
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6547 </head> |
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6548 <pb file="add_6787_f467v" o="467v" n="933"/> |
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6549 <pb file="add_6787_f468" o="468" n="934"/> |
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6550 <pb file="add_6787_f468v" o="468v" n="935"/> |
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6551 <pb file="add_6787_f469" o="469" n="936"/> |
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6552 <pb file="add_6787_f469v" o="469v" n="937"/> |
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6553 <pb file="add_6787_f470" o="470" n="938"/> |
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6554 <pb file="add_6787_f470v" o="470v" n="939"/> |
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6555 <pb file="add_6787_f471" o="471" n="940"/> |
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6556 <pb file="add_6787_f471v" o="471v" n="941"/> |
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6557 <pb file="add_6787_f472" o="472" n="942"/> |
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6558 <pb file="add_6787_f472v" o="472v" n="943"/> |
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6559 <pb file="add_6787_f473" o="473" n="944"/> |
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6560 <pb file="add_6787_f473v" o="473v" n="945"/> |
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6561 <pb file="add_6787_f474" o="474" n="946"/> |
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6562 <pb file="add_6787_f474v" o="474v" n="947"/> |
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6563 <pb file="add_6787_f475" o="475" n="948"/> |
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6564 <div xml:id="echoid-div129" type="page_commentary" level="2" n="129"> |
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6565 <p> |
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6566 <s xml:id="echoid-s587" xml:space="preserve">[<emph style="it">Note: |
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6567 <p> |
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6568 <s xml:id="echoid-s587" xml:space="preserve"> |
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6569 The references on this page are to Francesco Barozzi, |
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6570 <emph style="it">Admirandum illud geometricum problema tredecim modis demonstratum </emph> (1586) |
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6571 and to Federico Commandino, |
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6572 <emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566). <lb/> |
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6573 Pages 98 and 52 of the <emph style="it">Amirandum</emph> both contain diagrams. |
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6574 Harriot's (lower case) letters correspond to the (upper case) letters in Barozzi's diagrams. <lb/> |
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6575 The proposition of page 14 of <emph style="it">Conicorum libri quattuor</emph> is Proposition 11, |
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6576 Apollonius's definition of a parabola. For the full statement of this proposition see Add MS 6787, f. 357. |
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6577 </s> |
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6578 </p> |
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6579 </emph>] |
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6580 <lb/><lb/></s></p></div> |
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6581 <head xml:id="echoid-head124" xml:space="preserve" xml:lang="lat"> |
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6582 Barocius. pag. 98. |
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6583 </head> |
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6584 <p xml:lang="lat"> |
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6585 <s xml:id="echoid-s589" xml:space="preserve"> |
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6586 Hyperbola |
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6587 </s> |
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6588 </p> |
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6589 <p xml:lang="lat"> |
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6590 <s xml:id="echoid-s590" xml:space="preserve"> |
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6591 Barocius <lb/> |
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6592 pag. 52. |
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6593 </s> |
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6594 </p> |
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6595 <p xml:lang="lat"> |
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6596 <s xml:id="echoid-s591" xml:space="preserve"> |
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6597 Appol. pag. 14. parabola |
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6598 <lb/>[<emph style="it">tr: |
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6599 Apollonius, page 14, parabola |
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6600 </emph>]<lb/> |
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6601 </s> |
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6602 </p> |
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6603 <p xml:lang="lat"> |
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6604 <s xml:id="echoid-s592" xml:space="preserve"> |
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6605 Brevissime |
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6606 <lb/>[<emph style="it">tr: |
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6607 very briefly |
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6608 </emph>]<lb/> |
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6609 </s> |
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6610 </p> |
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6611 <pb file="add_6787_f475v" o="475v" n="949"/> |
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6612 <pb file="add_6787_f476" o="476" n="950"/> |
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6613 <p> |
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6614 <s xml:id="echoid-s593" xml:space="preserve"> |
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6615 <sc> |
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6616 A span is approximately 8 inches, or 20 cm. |
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6617 </sc> |
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6618 25. spannes is a pole in length. |
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6619 </s> |
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6620 <lb/> |
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6621 <s xml:id="echoid-s594" xml:space="preserve"> |
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6622 625. square spannes is a pole square. |
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6623 </s> |
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6624 <lb/> |
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6625 <s xml:id="echoid-s595" xml:space="preserve"> |
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6626 1250. spannes is 2 square poles <emph style="st">square</emph> |
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6627 </s> |
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6628 </p> |
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6629 <p> |
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6630 <s xml:id="echoid-s596" xml:space="preserve"> |
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6631 <sc> |
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6632 There are 4 poles to one chain, so 16 square poles to one square chain. |
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6633 </sc> |
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6634 or a square of the <lb/> |
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6635 whole [¿]chayne[?] |
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6636 </s> |
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6637 <lb/> |
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6638 <s xml:id="echoid-s597" xml:space="preserve"> |
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6639 <sc> |
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6640 There are 10 square chains to an acre. |
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6641 </sc> |
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6642 or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>0</mn></mrow></mfrac></mstyle></math> of an acre |
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6643 </s> |
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6644 <lb/> |
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6645 <s xml:id="echoid-s598" xml:space="preserve"> |
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6646 or: a day worke. |
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6647 </s> |
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6648 </p> |
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6649 <pb file="add_6787_f476v" o="476v" n="951"/> |
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6650 <pb file="add_6787_f477" o="477" n="952"/> |
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6651 <pb file="add_6787_f477v" o="477v" n="953"/> |
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6652 <pb file="add_6787_f478" o="478" n="954"/> |
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6653 <pb file="add_6787_f478v" o="478v" n="955"/> |
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6654 <pb file="add_6787_f479" o="479" n="956"/> |
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6655 <pb file="add_6787_f479v" o="479v" n="957"/> |
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parents:
diff
changeset
|
6656 <pb file="add_6787_f480" o="480" n="958"/> |
|
22d6a63640c6
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casties
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changeset
|
6657 <pb file="add_6787_f480v" o="480v" n="959"/> |
|
22d6a63640c6
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casties
parents:
diff
changeset
|
6658 <pb file="add_6787_f481" o="481" n="960"/> |
|
22d6a63640c6
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casties
parents:
diff
changeset
|
6659 <pb file="add_6787_f481v" o="481v" n="961"/> |
|
22d6a63640c6
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casties
parents:
diff
changeset
|
6660 <pb file="add_6787_f482" o="482" n="962"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6661 <pb file="add_6787_f482v" o="482v" n="963"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6662 <pb file="add_6787_f483" o="483" n="964"/> |
|
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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|
6663 <pb file="add_6787_f483v" o="483v" n="965"/> |
|
22d6a63640c6
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casties
parents:
diff
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|
6664 <pb file="add_6787_f484" o="484" n="966"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6665 <pb file="add_6787_f484v" o="484v" n="967"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6666 <pb file="add_6787_f485" o="485" n="968"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6667 <pb file="add_6787_f485v" o="485v" n="969"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6668 <pb file="add_6787_f486" o="486" n="970"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6669 <pb file="add_6787_f486v" o="486v" n="971"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6670 <pb file="add_6787_f487" o="487" n="972"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6671 <pb file="add_6787_f487v" o="487v" n="973"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6672 <pb file="add_6787_f488" o="488" n="974"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6673 <pb file="add_6787_f488v" o="488v" n="975"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6674 <pb file="add_6787_f489" o="489" n="976"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6675 <pb file="add_6787_f489v" o="489v" n="977"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6676 <pb file="add_6787_f490" o="490" n="978"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6677 <pb file="add_6787_f490v" o="490v" n="979"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6678 <pb file="add_6787_f491" o="491" n="980"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6679 <pb file="add_6787_f491v" o="491v" n="981"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6680 <pb file="add_6787_f492" o="492" n="982"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6681 <pb file="add_6787_f492v" o="492v" n="983"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6682 <pb file="add_6787_f493" o="493" n="984"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6683 <pb file="add_6787_f493v" o="493v" n="985"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6684 <pb file="add_6787_f494" o="494" n="986"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6685 <pb file="add_6787_f494v" o="494v" n="987"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6686 <pb file="add_6787_f495" o="495" n="988"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6687 <pb file="add_6787_f495v" o="495v" n="989"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6688 <p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6689 <s xml:id="echoid-s599" 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
|
6690 Hilles. a joyner. to cutte the parabola <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6691 at Ponden in Essex by Walton abbey <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6692 Well. Borne. |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6693 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6694 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6695 <p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6696 <s xml:id="echoid-s600" 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
|
6697 Nic. Burcot <emph style="super">Burked</emph> a turner <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6698 of town or shire <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6699 of Chisleworth <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6700 mint miller <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6701 In St Annes Lane <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6702 within Aldersgate |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6703 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6704 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6705 <pb file="add_6787_f496" o="496" n="990"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6706 <pb file="add_6787_f496v" o="496v" n="991"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6707 <pb file="add_6787_f497" o="497" n="992"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6708 <pb file="add_6787_f497v" o="497v" n="993"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6709 <pb file="add_6787_f498" o="498" n="994"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6710 <pb file="add_6787_f498v" o="498v" n="995"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6711 <pb file="add_6787_f499" o="499" n="996"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6712 <pb file="add_6787_f499v" o="499v" n="997"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6713 <pb file="add_6787_f500" o="500" n="998"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6714 <pb file="add_6787_f500v" o="500v" n="999"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6715 <pb file="add_6787_f501" o="501" n="1000"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6716 <pb file="add_6787_f501v" o="501v" n="1001"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6717 <pb file="add_6787_f502" o="502" n="1002"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6718 <pb file="add_6787_f502v" o="502v" n="1003"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6719 <pb file="add_6787_f503" o="503" n="1004"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6720 <pb file="add_6787_f503v" o="503v" n="1005"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6721 <pb file="add_6787_f504" o="504" n="1006"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6722 <pb file="add_6787_f504v" o="504v" n="1007"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6723 <pb file="add_6787_f505" o="505" n="1008"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6724 <pb file="add_6787_f505v" o="505v" n="1009"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6725 <pb file="add_6787_f506" o="506" n="1010"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6726 <pb file="add_6787_f506v" o="506v" n="1011"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6727 <pb file="add_6787_f507" o="507" n="1012"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6728 <pb file="add_6787_f507v" o="507v" n="1013"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6729 <pb file="add_6787_f508" o="508" n="1014"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6730 <pb file="add_6787_f508v" o="508v" n="1015"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6731 <pb file="add_6787_f509" o="509" n="1016"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6732 <pb file="add_6787_f509v" o="509v" n="1017"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6733 <pb file="add_6787_f510" o="510" n="1018"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6734 <pb file="add_6787_f510v" o="510v" n="1019"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6735 <pb file="add_6787_f511" o="511" n="1020"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6736 <pb file="add_6787_f511v" o="511v" n="1021"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6737 <pb file="add_6787_f512" o="512" n="1022"/> |
|
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6738 <pb file="add_6787_f512v" o="512v" n="1023"/> |
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|
6739 <pb file="add_6787_f513" o="513" n="1024"/> |
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|
6740 <pb file="add_6787_f513v" o="513v" n="1025"/> |
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|
6741 <pb file="add_6787_f514" o="514" n="1026"/> |
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|
6742 <pb file="add_6787_f514v" o="514v" n="1027"/> |
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|
6743 <pb file="add_6787_f515" o="515" n="1028"/> |
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|
6744 <pb file="add_6787_f515v" o="515v" n="1029"/> |
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|
6745 <pb file="add_6787_f516" o="516" n="1030"/> |
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|
6746 <pb file="add_6787_f516v" o="516v" n="1031"/> |
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|
6747 <pb file="add_6787_f517" o="517" n="1032"/> |
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|
6748 <pb file="add_6787_f517v" o="517v" n="1033"/> |
|
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|
6749 <pb file="add_6787_f518" o="518" n="1034"/> |
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|
6750 <pb file="add_6787_f518v" o="518v" n="1035"/> |
|
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|
6751 <pb file="add_6787_f519" o="519" n="1036"/> |
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|
6752 <pb file="add_6787_f519v" o="519v" n="1037"/> |
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|
6753 <pb file="add_6787_f520" o="520" n="1038"/> |
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|
6754 <pb file="add_6787_f520v" o="520v" n="1039"/> |
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|
6755 <pb file="add_6787_f521" o="521" n="1040"/> |
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|
6756 <pb file="add_6787_f521v" o="521v" n="1041"/> |
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|
6757 <pb file="add_6787_f522" o="522" n="1042"/> |
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|
6758 <pb file="add_6787_f522v" o="522v" n="1043"/> |
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|
6759 <pb file="add_6787_f523" o="523" n="1044"/> |
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|
6760 <pb file="add_6787_f523v" o="523v" n="1045"/> |
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|
6761 <pb file="add_6787_f524" o="524" n="1046"/> |
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|
6762 <pb file="add_6787_f524v" o="524v" n="1047"/> |
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|
6763 <pb file="add_6787_f525" o="525" n="1048"/> |
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|
6764 <div xml:id="echoid-div130" type="page_commentary" level="2" n="130"> |
|
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|
6765 <p> |
|
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|
6766 <s xml:id="echoid-s601" xml:space="preserve">[<emph style="it">Note: |
|
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|
6767 <p> |
|
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|
6768 <s xml:id="echoid-s601" xml:space="preserve"> |
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|
6769 This page refers to Proposition I.34 of Apollonius, as edited by Commandino in |
|
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|
6770 <emph style="it">Conicorum libri quattuor</emph> (1566). |
|
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|
6771 </s> |
|
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|
6772 <lb/> |
|
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|
6773 <quote> |
|
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|
6774 I.34 If on a hyperbola or ellipse or circumference of a circle some point is taken, |
|
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|
6775 and from it a straight line is dropped as ordinate to the diameter, |
|
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|
6776 and if the straight lines which the ordinate cuts off from the ends of the figure 19s transverse side |
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|
6777 have to each other a ratio which other segments of the transverse side have to each other, |
|
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|
6778 so that the segments from the vertex are corresponding, then the straight line joining the point |
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|
6779 taken on the transverse side and that taken on the section will touch the section. |
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|
6780 </quote> |
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|
6781 </p> |
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|
6782 </emph>] |
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|
6783 <lb/><lb/></s></p></div> |
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|
6784 <head xml:id="echoid-head125" xml:space="preserve" xml:lang="lat"> |
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|
6785 ut in elementa <lb/> |
|
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|
6786 conico ad tactus <lb/> |
|
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|
6787 prop. 34. lib. 1. <lb/> |
|
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|
6788 Appol. |
|
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|
6789 <lb/>[<emph style="it">tr: |
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|
6790 as in the elements of tangents to a cone, Proposition 34, Book 1, of Apollonius. |
|
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|
6791 </emph>]<lb/> |
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|
6792 </head> |
|
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|
6793 <pb file="add_6787_f525v" o="525v" n="1049"/> |
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|
6794 <pb file="add_6787_f526" o="526" n="1050"/> |
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|
6795 <pb file="add_6787_f526v" o="526v" n="1051"/> |
|
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|
6796 <pb file="add_6787_f527" o="527" n="1052"/> |
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|
6797 <pb file="add_6787_f527v" o="527v" n="1053"/> |
|
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|
6798 <pb file="add_6787_f528" o="528" n="1054"/> |
|
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|
6799 <pb file="add_6787_f528v" o="528v" n="1055"/> |
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|
6800 <pb file="add_6787_f529" o="529" n="1056"/> |
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|
6801 <pb file="add_6787_f529v" o="529v" n="1057"/> |
|
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|
6802 <pb file="add_6787_f530" o="530" n="1058"/> |
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|
6803 <pb file="add_6787_f530v" o="530v" n="1059"/> |
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|
6804 <pb file="add_6787_f531" o="531" n="1060"/> |
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|
6805 <pb file="add_6787_f531v" o="531v" n="1061"/> |
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|
6806 <pb file="add_6787_f532" o="532" n="1062"/> |
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|
6807 <pb file="add_6787_f532v" o="532v" n="1063"/> |
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|
6808 <pb file="add_6787_f533" o="533" n="1064"/> |
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|
6809 <pb file="add_6787_f533v" o="533v" n="1065"/> |
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|
6810 <pb file="add_6787_f534" o="534" n="1066"/> |
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6811 <pb file="add_6787_f534v" o="534v" n="1067"/> |
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6812 <pb file="add_6787_f535" o="535" n="1068"/> |
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|
6813 <pb file="add_6787_f535v" o="535v" n="1069"/> |
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|
6814 <pb file="add_6787_f536" o="536" n="1070"/> |
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|
6815 <pb file="add_6787_f536v" o="536v" n="1071"/> |
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|
6816 <pb file="add_6787_f537" o="537" n="1072"/> |
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|
6817 <div xml:id="echoid-div131" type="page_commentary" level="2" n="131"> |
|
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|
6818 <p> |
|
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|
6819 <s xml:id="echoid-s603" xml:space="preserve">[<emph style="it">Note: |
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6820 <p> |
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6821 <s xml:id="echoid-s603" xml:space="preserve"> |
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6822 This page refers to Propositions I.21 and II.1 of Apollonius, as edited by Commandino in |
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6823 <emph style="it">Conicorum libri quattuor</emph> (1566). |
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6824 </s> |
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6825 <lb/> |
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6826 <quote> |
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6827 I.21 |
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6828 If in a hyperbola or ellipse or circumference of a circle straight lines are dropped as ordinates to the diameter, |
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6829 the square on them will be to the areas contained by the straight lines cut off by them beginning from the ends of |
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6830 the transverse side of the figure, as the upright side of the figure is to the transverse, |
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6831 and to each other as the areas contained by the straight lines cut off, as we have said. |
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6832 </quote> |
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6833 <lb/> |
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6834 <quote> |
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6835 II.1 |
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6836 If a straight line touch an hyperbola at its vertex, and from it on both sides of the diamter |
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6837 a straight line is cut off equal in square to the fourth of the figure, |
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6838 then the straight lines drawn from the centre of the section to the ends thus taken on the tangent |
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6839 will not meet the section. |
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6840 </quote> |
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6841 </p> |
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6842 </emph>] |
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6843 <lb/><lb/></s></p></div> |
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6844 <head xml:id="echoid-head126" xml:space="preserve" xml:lang="lat"> |
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6845 B.2. Appol. lib. 1. pr. 21 De hyperbola |
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6846 <lb/>[<emph style="it">tr: |
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6847 Apollonius, Book I, Proposition 21. On the hyperbola. |
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6848 </emph>]<lb/> |
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6849 </head> |
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6850 <p xml:lang="lat"> |
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6851 <s xml:id="echoid-s605" xml:space="preserve"> |
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6852 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>x</mi></mstyle></math> latus <lb/> |
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6853 rectum. |
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6854 <lb/>[<emph style="it">tr: |
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6855 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>x</mi></mstyle></math> is the latus rectum |
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6856 </emph>]<lb/> |
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6857 </s> |
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6858 <lb/> |
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6859 <s xml:id="echoid-s606" xml:space="preserve"> |
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6860 2<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>u</mi></mstyle></math> = diametro 2<emph style="super">a</emph> |
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6861 <lb/>[<emph style="it">tr: |
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6862 2<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>u</mi></mstyle></math> is the second diameter |
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6863 </emph>]<lb/> |
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6864 </s> |
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6865 <lb/> |
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6866 <s xml:id="echoid-s607" xml:space="preserve"> |
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6867 [<emph style="it">Note: |
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6868 For two nearby sheets labelled B.4 see Add MS 6787, f. 545, f. 546. |
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6869 </emph>]<lb/> |
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6870 si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>u</mi></mstyle></math> asymptotos. Ut est per B.4 <lb/> |
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6871 per 1p. 2, lib: Apol. |
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6872 <lb/>[<emph style="it">tr: |
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6873 If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>u</mi></mstyle></math> is an asymptote, as it is by B.4., then by Proposition 1, Book 2 of Apollonius |
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6874 </emph>]<lb/> |
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6875 </s> |
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6876 </p> |
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6877 <pb file="add_6787_f537v" o="537v" n="1073"/> |
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6878 <pb file="add_6787_f538" o="538" n="1074"/> |
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6879 <pb file="add_6787_f538v" o="538v" n="1075"/> |
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6880 <pb file="add_6787_f539" o="539" n="1076"/> |
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6881 <pb file="add_6787_f539v" o="539v" n="1077"/> |
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6882 <pb file="add_6787_f540" o="540" n="1078"/> |
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6883 <pb file="add_6787_f540v" o="540v" n="1079"/> |
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6884 <pb file="add_6787_f541" o="541" n="1080"/> |
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6885 <pb file="add_6787_f541v" o="541v" n="1081"/> |
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6886 <pb file="add_6787_f542" o="542" n="1082"/> |
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6887 <pb file="add_6787_f542v" o="542v" n="1083"/> |
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6888 <pb file="add_6787_f543" o="543" n="1084"/> |
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6889 <pb file="add_6787_f543v" o="543v" n="1085"/> |
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6890 <pb file="add_6787_f544" o="544" n="1086"/> |
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6891 <pb file="add_6787_f544v" o="544v" n="1087"/> |
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6892 <pb file="add_6787_f545" o="545" n="1088"/> |
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6893 <pb file="add_6787_f545v" o="545v" n="1089"/> |
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6894 <pb file="add_6787_f546" o="546" n="1090"/> |
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6895 <pb file="add_6787_f546v" o="546v" n="1091"/> |
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6896 <pb file="add_6787_f547" o="547" n="1092"/> |
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6897 <pb file="add_6787_f547v" o="547v" n="1093"/> |
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6898 <pb file="add_6787_f548" o="548" n="1094"/> |
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6899 <pb file="add_6787_f548v" o="548v" n="1095"/> |
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6900 <pb file="add_6787_f549" o="549" n="1096"/> |
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6901 <div xml:id="echoid-div132" type="page_commentary" level="2" n="132"> |
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6902 <p> |
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changeset
|
6903 <s xml:id="echoid-s608" 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/
casties
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diff
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|
6904 <p> |
|
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
|
6905 <s xml:id="echoid-s608" 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
|
6906 The references on this page are to the verses of Leviticus, Chapter 18. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6907 The rules in the uper left column are taken directly from the verses of Leviticus, |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6908 and refer to sexual relations prohibited to a man. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6909 In the upper right column, Harriot has given the equivalent relations for a woman. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6910 In the lower columns, Harriot has reversed the generations, |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6911 so that 'father' is replaced by 'son', 'mother' by 'daughter', and so on. |
|
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
|
6912 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6913 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6914 </emph>] |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6915 <lb/><lb/></s></p></div> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6916 <head xml:id="echoid-head127" 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
|
6917 Leviticus. cap. 18 |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6918 </head> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6919 <p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6920 <s xml:id="echoid-s610" 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
|
6921 A man shall not <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6922 marry. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6923 7) <emph style="super">verse</emph> his mother <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6924 8. his fathers wife <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6925 or stepmother <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6926 9. <emph style="super">his sister being:</emph> the daughter of his father <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6927 or the daughter of his mother. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6928 10. his sonnes daughter. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6929 or his daughters daughter. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6930 11. his fathers wifes daughter, <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6931 begotten by the father. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6932 12. his fathers sister. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6933 13. his mothers sister. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6934 14. his fathers brothers wife. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6935 15. his sonnes wife. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6936 16. his brothers wife. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6937 17. his wives daughter. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6938 (his wives daughters daughter) <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6939 (his wives sonnes daughter). |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6940 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6941 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6942 <p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6943 <s xml:id="echoid-s611" 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
|
6944 A woman shall <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6945 not marry. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6946 7) <emph style="super">verse</emph> her father <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6947 8. her mothers hsuband <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6948 or stepmother <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6949 9. her brother being: <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6950 the sonne of her father <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6951 or the sonne of her mother. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6952 10. her daughters sonne. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6953 or her sonnes sonne. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6954 11. her mothers husbands sonne. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6955 12. her mothers brother. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6956 13. her fathers brother. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6957 14. her mothers sisters husband. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6958 15. her daughters hsuband. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6959 16. her sisters husband. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6960 17. her husbands sonne. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6961 (her husbands sonnes sonne) <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6962 (her husbands daughters sonne). |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6963 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6964 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6965 <p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6966 <s xml:id="echoid-s612" 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
|
6967 A woman shall <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6968 not marry. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6969 7) <emph style="super">verse</emph> her sonne <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6970 8. her husbandes sonne <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6971 or stepmother <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6972 9. her fathers sonne <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6973 her mothers sonne <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6974 10. her fathers father. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6975 her mothers mother. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6976 11. her mothers husbands sonne. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6977 12. her brothers sonne. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6978 13. her sisters sonne. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6979 14. her husbands brothers sonne. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6980 15. her hsubands father. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6981 16. her husbands brother. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6982 17. her mothers husband. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6983 (her mothers mothers hsuband) <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6984 (her fathers mothesr husband). |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6985 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6986 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6987 <p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6988 <s xml:id="echoid-s613" 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
|
6989 A man shall <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6990 not marry. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6991 7) <emph style="super">verse</emph> his daughter <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6992 8. his wives daughter <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
6993 9. his father daughter. <lb/> |
|
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|
6994 his mothers daughter. <lb/> |
|
22d6a63640c6
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|
6995 10. his mothers mother. <lb/> |
|
22d6a63640c6
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|
6996 his fathers mother. <lb/> |
|
22d6a63640c6
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|
6997 11. his father wives daughter. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
6998 12. his sisters daughter. <lb/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
6999 13. his brothers daughter. <lb/> |
|
22d6a63640c6
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|
7000 14. his wives sisters daughter. <lb/> |
|
22d6a63640c6
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diff
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|
7001 15. his wives <emph style="st">daughter</emph> mother. <lb/> |
|
22d6a63640c6
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|
7002 16. his wives sister. <lb/> |
|
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|
7003 17. his fathers wife. <lb/> |
|
22d6a63640c6
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|
7004 (his fathers fathers wife) <lb/> |
|
22d6a63640c6
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diff
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|
7005 (his mothers fathers wife). |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
7006 </s> |
|
22d6a63640c6
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|
7007 </p> |
|
22d6a63640c6
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|
7008 <pb file="add_6787_f549v" o="549v" n="1097"/> |
|
22d6a63640c6
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|
7009 <pb file="add_6787_f550" o="550" n="1098"/> |
|
22d6a63640c6
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|
7010 <pb file="add_6787_f550v" o="550v" n="1099"/> |
|
22d6a63640c6
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|
7011 <pb file="add_6787_f551" o="551" n="1100"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
7012 <pb file="add_6787_f551v" o="551v" n="1101"/> |
|
22d6a63640c6
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|
7013 <pb file="add_6787_f552" o="552" n="1102"/> |
|
22d6a63640c6
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|
7014 <pb file="add_6787_f552v" o="552v" n="1103"/> |
|
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|
7015 <pb file="add_6787_f553" o="553" n="1104"/> |
|
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|
7016 <pb file="add_6787_f553v" o="553v" n="1105"/> |
|
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|
7017 <pb file="add_6787_f554" o="554" n="1106"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
7018 <pb file="add_6787_f554v" o="554v" n="1107"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
7019 <pb file="add_6787_f555" o="555" n="1108"/> |
|
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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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|
7020 <pb file="add_6787_f555v" o="555v" n="1109"/> |
|
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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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|
7021 <pb file="add_6787_f556" o="556" n="1110"/> |
|
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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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|
7022 <pb file="add_6787_f556v" o="556v" n="1111"/> |
|
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|
7023 <pb file="add_6787_f557" o="557" n="1112"/> |
|
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|
7024 <pb file="add_6787_f557v" o="557v" n="1113"/> |
|
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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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|
7025 <pb file="add_6787_f558" o="558" n="1114"/> |
|
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|
7026 <pb file="add_6787_f558v" o="558v" n="1115"/> |
|
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|
7027 <pb file="add_6787_f559" o="559" n="1116"/> |
|
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|
7028 <pb file="add_6787_f559v" o="559v" n="1117"/> |
|
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|
7029 <pb file="add_6787_f560" o="560" n="1118"/> |
|
22d6a63640c6
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|
7030 <pb file="add_6787_f560v" o="560v" n="1119"/> |
|
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|
7031 <pb file="add_6787_f561" o="561" n="1120"/> |
|
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|
7032 <pb file="add_6787_f561v" o="561v" n="1121"/> |
|
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|
7033 <pb file="add_6787_f562" o="562" n="1122"/> |
|
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|
7034 <pb file="add_6787_f562v" o="562v" n="1123"/> |
|
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|
7035 <pb file="add_6787_f563" o="563" n="1124"/> |
|
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|
7036 <pb file="add_6787_f563v" o="563v" n="1125"/> |
|
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|
7037 <pb file="add_6787_f564" o="564" n="1126"/> |
|
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|
7038 <pb file="add_6787_f564v" o="564v" n="1127"/> |
|
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|
7039 <pb file="add_6787_f565" o="565" n="1128"/> |
|
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|
7040 <p> |
|
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|
7041 <s xml:id="echoid-s614" xml:space="preserve"> |
|
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|
7042 a.b.c.d. ... <lb/> |
|
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|
7043 A.B.C.D. ... |
|
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|
7044 </s> |
|
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|
7045 </p> |
|
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|
7046 <pb file="add_6787_f565v" o="565v" n="1129"/> |
|
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|
7047 <pb file="add_6787_f566" o="566" n="1130"/> |
|
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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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parents:
diff
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|
7048 <div xml:id="echoid-div133" type="page_commentary" level="2" n="133"> |
|
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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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|
7049 <p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff
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|
7050 <s xml:id="echoid-s615" xml:space="preserve">[<emph style="it">Note: |
|
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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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parents:
diff
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|
7051 <p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
7052 <s xml:id="echoid-s615" 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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parents:
diff
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|
7053 The reference to Viète is to his |
|
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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|
7054 <emph style="it">Variorum responsorum liber VIII</emph>, |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff
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|
7055 Chapter 17, entitled 'Progressio geometrica' (see also Add MS 6786, f. 453v), and |
|
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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|
7056 Chapter 18, entitled 'Polygonorum circulo ordinate inscriptorum ad circulum ratio'. |
|
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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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|
7057 </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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|
7058 </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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|
7059 </emph>] |
|
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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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parents:
diff
changeset
|
7060 <lb/><lb/></s></p></div> |
|
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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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|
7061 <head xml:id="echoid-head128" 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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|
7062 Some considerations rising upon the 23. p. of <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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|
7063 Archimedes De Quadratura parabolæ & the 17 & 18 chap. of <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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|
7064 Viætus his responsorum. pa. 29. |
|
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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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|
7065 </head> |
|
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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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|
7066 <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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|
7067 <s xml:id="echoid-s617" 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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|
7068 If there be nombers in subduple proportion infinitely to find out <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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|
7069 their summe. |
|
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|
7070 </s> |
|
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parents:
diff
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|
7071 <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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|
7072 <s xml:id="echoid-s618" 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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|
7073 as 8.4.2.1.<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>.<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>.<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>8</mn></mrow></mfrac></mstyle></math>.<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>6</mn></mrow></mfrac></mstyle></math>.&c. |
|
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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
|
7074 </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff
changeset
|
7075 <lb/> |
|
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|
7076 <s xml:id="echoid-s619" xml:space="preserve"> |
|
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diff
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|
7077 The first doubled is the summe. that is in this example .8. doubled <emph style="st">is</emph> is 16. |
|
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|
7078 </s> |
|
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diff
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|
7079 <lb/> |
|
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casties
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|
7080 <s xml:id="echoid-s620" xml:space="preserve"> |
|
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casties
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diff
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|
7081 In this <emph style="super">kind of</emph> progression if the number of places be finite. the summe is found <lb/> |
|
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casties
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|
7082 thus. <emph style="st">waye</emph>. |
|
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|
7083 </s> |
|
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|
7084 <s xml:id="echoid-s621" xml:space="preserve"> |
|
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casties
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diff
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|
7085 Double the first as here .8. which maketh 16. from it subtract the <lb/> |
|
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|
7086 last & the remayne wilbe the summe. |
|
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|
7087 </s> |
|
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|
7088 <s xml:id="echoid-s622" xml:space="preserve"> |
|
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diff
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|
7089 for alwayes this <emph style="it">last</emph> doth lacke of <lb/> |
|
22d6a63640c6
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|
7090 the double of the first [???] <emph style="super">the</emph> quantity of the last. |
|
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diff
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|
7091 </s> |
|
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diff
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|
7092 <lb/> |
|
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|
7093 <s xml:id="echoid-s623" xml:space="preserve"> |
|
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|
7094 all the quantityes after the first, are æquall to the first wanting <lb/> |
|
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|
7095 the quantity of the last. |
|
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|
7096 </s> |
|
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diff
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|
7097 <s xml:id="echoid-s624" xml:space="preserve"> |
|
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|
7098 as 4 & 2 are æquall to 8 wanting 2. <lb/> |
|
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|
7099 4.2.1. are æquall wanting one. 4.2.1.<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>.<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>. are æquall wanting <lb/> |
|
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diff
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|
7100 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>, & so forth. |
|
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|
7101 </s> |
|
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diff
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|
7102 <lb/> |
|
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|
7103 <s xml:id="echoid-s625" xml:space="preserve"> |
|
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diff
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|
7104 The summe of that whole progression the first being 8. & the last <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>6</mn></mrow></mfrac></mstyle></math> <lb/> |
|
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|
7105 thereby. |
|
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|
7106 </s> |
|
22d6a63640c6
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diff
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|
7107 <lb/> |
|
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|
7108 <s xml:id="echoid-s626" xml:space="preserve"> |
|
22d6a63640c6
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diff
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|
7109 All the fractions under .1. if they <lb/> |
|
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|
7110 be of a finite number of places <lb/> |
|
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|
7111 are lesse than .1. <lb/> |
|
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|
7112 if infinite æquall to .1. |
|
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|
7113 </s> |
|
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diff
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|
7114 </p> |
|
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diff
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|
7115 <p> |
|
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|
7116 <s xml:id="echoid-s627" xml:space="preserve"> |
|
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diff
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|
7117 Demonstration |
|
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|
7118 </s> |
|
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|
7119 <lb/> |
|
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|
7120 <s xml:id="echoid-s628" xml:space="preserve"> |
|
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|
7121 They are neither greater nor lesse. |
|
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|
7122 </s> |
|
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diff
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|
7123 <lb/> |
|
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|
7124 <s xml:id="echoid-s629" xml:space="preserve"> |
|
22d6a63640c6
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diff
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|
7125 If greater, it is more than the progression; for the summe of all <lb/> |
|
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|
7126 after the first, must be lesse than the first by the quantity of the last. |
|
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|
7127 </s> |
|
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diff
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|
7128 <lb/> |
|
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diff
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|
7129 <s xml:id="echoid-s630" xml:space="preserve"> |
|
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diff
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|
7130 If lesse, then <emph style="st">the last must</emph> must it be lesse <emph style="st">th</emph> by a quantity <lb/> |
|
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|
7131 & the progression is not yet ended but it is supposed infinite. |
|
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|
7132 </s> |
|
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diff
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|
7133 <lb/> |
|
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|
7134 <s xml:id="echoid-s631" xml:space="preserve"> |
|
22d6a63640c6
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casties
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diff
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|
7135 Therefore &c. |
|
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diff
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|
7136 </s> |
|
22d6a63640c6
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diff
changeset
|
7137 </p> |
|
22d6a63640c6
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diff
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|
7138 <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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|
7139 <s xml:id="echoid-s632" 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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|
7140 In lines thus. |
|
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changeset
|
7141 </s> |
|
22d6a63640c6
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diff
changeset
|
7142 </p> |
|
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diff
changeset
|
7143 <p> |
|
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|
7144 <s xml:id="echoid-s633" 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
changeset
|
7145 The demonstration of all <lb/> |
|
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diff
changeset
|
7146 is, it cannot be greater or <lb/> |
|
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diff
changeset
|
7147 lesse according to the <lb/> |
|
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casties
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diff
changeset
|
7148 condition and property of <lb/> |
|
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diff
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|
7149 the progression argued. |
|
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changeset
|
7150 </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
|
7151 </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
changeset
|
7152 <p> |
|
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changeset
|
7153 <s xml:id="echoid-s634" 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
changeset
|
7154 The like waye in playnes or bodyes. |
|
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changeset
|
7155 </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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diff
changeset
|
7156 </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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diff
changeset
|
7157 <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
|
7158 <s xml:id="echoid-s635" 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
changeset
|
7159 All fractions under <emph style="super">& from</emph> a unite. |
|
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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
|
7160 </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
|
7161 <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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changeset
|
7162 <s xml:id="echoid-s636" xml:space="preserve"> |
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7163 Subtriple are æquall to: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>. |
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7164 </s> |
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7165 <lb/> |
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7166 <s xml:id="echoid-s637" xml:space="preserve"> |
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7167 Subquadruple: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math>. |
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7168 </s> |
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7169 <lb/> |
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7170 <s xml:id="echoid-s638" xml:space="preserve"> |
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7171 Subquintuple: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>. |
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7172 </s> |
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7173 <lb/> |
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7174 <s xml:id="echoid-s639" xml:space="preserve"> |
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|
7175 &c. |
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7176 </s> |
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7177 </p> |
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7178 <p> |
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7179 <s xml:id="echoid-s640" xml:space="preserve"> |
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7180 Consider also of triangles inscribed from a square or triangles in a circle <lb/> |
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|
7181 how to get the sum of their progression ratione [???] |
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7182 </s> |
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7183 <lb/> |
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7184 <s xml:id="echoid-s641" xml:space="preserve"> |
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|
7185 Consider also what proportion or whether any <lb/> |
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|
7186 obtain for those above a unite & others &c. |
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7187 </s> |
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|
7188 </p> |
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|
7189 <p> |
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7190 <s xml:id="echoid-s642" xml:space="preserve"> |
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|
7191 see another paper |
|
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|
7192 </s> |
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|
7193 </p> |
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|
7194 <pb file="add_6787_f566v" o="566v" n="1131"/> |
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|
7195 <pb file="add_6787_f567" o="567" n="1132"/> |
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|
7196 <div xml:id="echoid-div134" type="page_commentary" level="2" n="134"> |
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|
7197 <p> |
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|
7198 <s xml:id="echoid-s643" xml:space="preserve">[<emph style="it">Note: |
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|
7199 <p> |
|
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|
7200 <s xml:id="echoid-s643" xml:space="preserve"> |
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|
7201 The terminology and examples on this page are taken from Viète, |
|
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|
7202 <emph style="it">In artem analyticen isagoge</emph>, 1591, Chapter V, |
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|
7203 Harriot has re-written Viète's examples in symbols, using <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo></mstyle></math> signs and |
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|
7204 the convention <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>A</mi></mstyle></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>-squared, and so on. |
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7205 The same material appears again on Add MS 6784, f. 325, but there in Harriot's more usual lower case notation. |
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|
7206 </s> |
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|
7207 </p> |
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|
7208 </emph>] |
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|
7209 <lb/><lb/></s></p></div> |
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|
7210 <p xml:lang="lat"> |
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|
7211 <s xml:id="echoid-s645" xml:space="preserve"> |
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|
7212 Antithesis. |
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|
7213 <lb/>[<emph style="it">tr: |
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|
7214 Antithesis |
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|
7215 </emph>]<lb/> |
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|
7216 </s> |
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|
7217 <lb/> |
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7218 <s xml:id="echoid-s646" xml:space="preserve"> |
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|
7219 ergo. |
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|
7220 <lb/>[<emph style="it">tr: |
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|
7221 therefore |
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|
7222 </emph>]<lb/> |
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|
7223 </s> |
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|
7224 <lb/> |
|
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|
7225 <s xml:id="echoid-s647" xml:space="preserve"> |
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|
7226 Ratio. |
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|
7227 <lb/>[<emph style="it">tr: |
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|
7228 Ratio |
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|
7229 </emph>]<lb/> |
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|
7230 </s> |
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|
7231 <lb/> |
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7232 <s xml:id="echoid-s648" xml:space="preserve"> |
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|
7233 adde: |
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|
7234 <lb/>[<emph style="it">tr: |
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|
7235 add |
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|
7236 </emph>]<lb/> |
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|
7237 </s> |
|
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|
7238 </p> |
|
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|
7239 <p xml:lang="lat"> |
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|
7240 <s xml:id="echoid-s649" xml:space="preserve"> |
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|
7241 Hypobibasmus. |
|
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|
7242 <lb/>[<emph style="it">tr: |
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|
7243 Hypobibasmus |
|
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|
7244 </emph>]<lb/> |
|
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|
7245 </s> |
|
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|
7246 <lb/> |
|
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|
7247 <s xml:id="echoid-s650" xml:space="preserve"> |
|
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|
7248 ergo: |
|
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|
7249 <lb/>[<emph style="it">tr: |
|
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|
7250 therefore: |
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7251 </emph>]<lb/> |
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|
7252 </s> |
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|
7253 </p> |
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7254 <p xml:lang="lat"> |
|
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|
7255 <s xml:id="echoid-s651" xml:space="preserve"> |
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|
7256 Parabolismus. |
|
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7257 <lb/>[<emph style="it">tr: |
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|
7258 Parabolismus |
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7259 </emph>]<lb/> |
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|
7260 </s> |
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|
7261 <lb/> |
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7262 <s xml:id="echoid-s652" xml:space="preserve"> |
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|
7263 ergo: |
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|
7264 <lb/>[<emph style="it">tr: |
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|
7265 therefore: |
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7266 </emph>]<lb/> |
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7267 </s> |
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7268 </p> |
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7269 <p xml:lang="lat"> |
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7270 <s xml:id="echoid-s653" xml:space="preserve"> |
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|
7271 Nota. ex nostra observatione |
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7272 <lb/>[<emph style="it">tr: |
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|
7273 Note, from my observation |
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7274 </emph>]<lb/> |
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7275 </s> |
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7276 <lb/> |
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7277 <s xml:id="echoid-s654" xml:space="preserve"> |
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7278 ergo per 14. p.6<emph style="super">i</emph>. vel 34. II<emph style="super">i</emph> |
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7279 <lb/>[<emph style="it">tr: |
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|
7280 therefore by page 14, proposition 6, or page 34, proposition 11 |
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|
7281 </emph>]<lb/> |
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7282 </s> |
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7283 <lb/> |
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7284 <s xml:id="echoid-s655" xml:space="preserve"> |
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|
7285 ergo |
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7286 <lb/>[<emph style="it">tr: |
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|
7287 therefore |
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|
7288 </emph>]<lb/> |
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|
7289 </s> |
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7290 <lb/> |
|
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7291 <s xml:id="echoid-s656" xml:space="preserve"> |
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|
7292 ergo etiam si <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>B</mi></mstyle></math> æqualia. |
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7293 <lb/>[<emph style="it">tr: |
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|
7294 therefore also if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> are equal. |
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7295 </emph>]<lb/> |
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7296 </s> |
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|
7297 </p> |
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|
7298 <pb file="add_6787_f567v" o="567v" n="1133"/> |
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|
7299 <pb file="add_6787_f568" o="568" n="1134"/> |
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|
7300 <div xml:id="echoid-div135" type="page_commentary" level="2" n="135"> |
|
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|
7301 <p> |
|
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|
7302 <s xml:id="echoid-s657" xml:space="preserve">[<emph style="it">Note: |
|
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|
7303 <p> |
|
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7304 <s xml:id="echoid-s657" xml:space="preserve"> |
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|
7305 This page refers to Proposition 53 from Book I of Apollonius, as edited by Commandino in |
|
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|
7306 <emph style="it">Conicorum libri quattuor</emph> (1566). |
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|
7307 Proposition (Problem) I.53 is an extension of Proposition (Problem) I.52. |
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|
7308 </s> |
|
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|
7309 <lb/> |
|
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|
7310 <quote> |
|
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|
7311 I.52. |
|
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|
7312 Given a straight line in a plane bounded at one point, to find in the plane the section of a cone |
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|
7313 called parabola, whose diameter is the given straight line, and whose vertex is the end of the straight line, |
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|
7314 and where whatever straight line is dropped from the section to the diameter at a given angle, |
|
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|
7315 will equal in square the rectangle contained by the straight line cut off by it from the vertex of the section |
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|
7316 and by some other given straight line. |
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|
7317 </quote> |
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|
7318 <lb/> |
|
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|
7319 <s xml:id="echoid-s658" xml:space="preserve"> |
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|
7320 There is also a reference to the equivalent construction for an ellipse, |
|
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|
7321 as demonstrated by Commandino in <emph style="it">Pappi Alexandrini mathematicae collectiones</emph> (1588). |
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|
7322 The construction, and Commandino's commentary on it, are to be found on pages 320v–321. |
|
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|
7323 </s> |
|
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|
7324 </p> |
|
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|
7325 </emph>] |
|
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|
7326 <lb/><lb/></s></p></div> |
|
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|
7327 <p xml:lang="lat"> |
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|
7328 <s xml:id="echoid-s660" xml:space="preserve"> |
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|
7329 Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> diameter <lb/> |
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|
7330 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> latus rectum <lb/> |
|
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|
7331 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi><mi>h</mi></mstyle></math> angulus ordin: applic: <lb/> |
|
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|
7332 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> centrum hyperboles <lb/> |
|
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|
7333 oportet invenire axem sectionis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>l</mi></mstyle></math> <lb/> |
|
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|
7334 et punctum verticis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi></mstyle></math>. |
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|
7335 <lb/>[<emph style="it">tr: |
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7336 according to Apollonius, Book 1, Proposition 53, page 38. |
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7337 </emph>]<lb/> |
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7338 </s> |
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7339 </p> |
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7340 <p xml:lang="lat"> |
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7341 <s xml:id="echoid-s661" xml:space="preserve"> |
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7342 1) fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>i</mi><mo>=</mo><mi>a</mi><mi>c</mi></mstyle></math> <lb/> |
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7343 et: periferia agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>i</mi></mstyle></math> <lb/> |
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7344 secabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> productam in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. <lb/> |
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7345 fiat: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mi>e</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> <lb/> |
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7346 et: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>q</mi><mo>=</mo><mi>a</mi><mi>p</mi></mstyle></math> <lb/> |
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7347 agantur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>q</mi></mstyle></math> productæ <lb/> |
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7348 et sunt asymptotæ hyperboles |
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7349 <lb/>[<emph style="it">tr: |
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7350 let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>i</mi><mo>=</mo><mi>a</mi><mi>c</mi></mstyle></math>, and take the circumference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>i</mi></mstyle></math>, which will cut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> produced at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>; |
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7351 let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mi>e</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>q</mi><mo>=</mo><mi>a</mi><mi>p</mi></mstyle></math>. |
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7352 Taking <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>q</mi></mstyle></math> produced, these will be the asymptotes of the hyperbolas. |
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7353 </emph>]<lb/> |
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7354 </s> |
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7355 </p> |
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7356 <p xml:lang="lat"> |
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7357 <s xml:id="echoid-s662" xml:space="preserve"> |
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7358 2<emph style="super">a</emph>) fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>r</mi><mo>=</mo><mi>d</mi><mi>q</mi></mstyle></math> <lb/> |
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7359 Agatur periferia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>s</mi><mi>p</mi></mstyle></math> <lb/> |
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7360 sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>s</mi></mstyle></math> ad angulos rectos <lb/> |
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7361 fiat: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>t</mi><mo>=</mo><mi>d</mi><mi>s</mi></mstyle></math> <lb/> |
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7362 et; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>u</mi><mo>=</mo><mi>d</mi><mi>t</mi></mstyle></math> <lb/> |
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7363 agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>t</mi></mstyle></math> <lb/> |
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7364 secetur ut per medium in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi></mstyle></math> <lb/> |
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7365 et fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>d</mi><mo>=</mo><mi>d</mi><mi>l</mi></mstyle></math> <lb/> |
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7366 Dico quod <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>l</mi></mstyle></math>, est axis <lb/> |
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7367 et puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi></mstyle></math>, est vertex sectionis. |
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7368 <lb/>[<emph style="it">tr: |
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7369 let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>r</mi><mo>=</mo><mi>d</mi><mi>q</mi></mstyle></math>; take the circumference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>s</mi><mi>p</mi></mstyle></math>, and set <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>s</mi></mstyle></math> at right angles. |
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7370 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>t</mi><mo>=</mo><mi>d</mi><mi>s</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>u</mi><mo>=</mo><mi>d</mi><mi>t</mi></mstyle></math>; taking <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>t</mi></mstyle></math>, it will be cut in the middle at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi></mstyle></math>; and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>d</mi><mo>=</mo><mi>d</mi><mi>l</mi></mstyle></math>. |
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7371 I say that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>l</mi></mstyle></math> is the axis, and the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi></mstyle></math> is the vertex of the section. |
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7372 </emph>]<lb/> |
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7373 </s> |
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7374 </p> |
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7375 <p xml:lang="lat"> |
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7376 <s xml:id="echoid-s663" xml:space="preserve"> |
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7377 cætera sunt de constructione <lb/> |
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7378 secundum Appol: lib. 1. pr: 53. pag: 38. <lb/> |
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7379 pro Elipsi <emph style="super">sequenti</emph> vide pappum, pag: 321 <lb/> |
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7380 et ibi Comandinum. Ubi nota quod ibi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>H</mi></mstyle></math> est <lb/> |
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7381 dimidium lateris rect. |
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7382 <lb/>[<emph style="it">tr: |
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7383 The rest concerns the construction according to Apollonius, Book 1, Proposition 53, page 38. |
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7384 For the ellipse following it, see Pappus, page 321, and that place in Commandinus, where he notes there that |
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7385 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>H</mi></mstyle></math> is half the latus rectum. |
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7386 </emph>]<lb/> |
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7387 </s> |
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7388 </p> |
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7389 <pb file="add_6787_f568v" o="568v" n="1135"/> |
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7390 <pb file="add_6787_f569" o="569" n="1136"/> |
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7391 <div xml:id="echoid-div136" type="page_commentary" level="2" n="136"> |
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7392 <p> |
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7393 <s xml:id="echoid-s664" xml:space="preserve">[<emph style="it">Note: |
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7394 <p> |
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7395 <s xml:id="echoid-s664" xml:space="preserve"> |
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7396 This page refers to Proposition 54 from Book I of Apollonius, as edited by Commandino in |
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7397 <emph style="it">Conicorum libri quattuor</emph> (1566). |
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7398 </s> |
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7399 <lb/> |
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7400 <quote> |
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7401 I.54. |
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7402 Given two bounded straight lines perpendicular to each other, one of them being produced |
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7403 on the side of the right angle, to find on the straight line produced the section of a cone |
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7404 called hyperbola in the same plane with the straight line, so that the straight line produced |
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|
7405 is a diameter of the section and the point at the angle is the vertex, and where whatever straight line |
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|
7406 is dropped from the section to the diameter, making an angle equal to a given angle, |
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|
7407 will equal in square the rectangle applied to the other straight line having as breadth |
|
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|
7408 the straight line cut off the dropped straight line beginning with the vertex and projecting beyond |
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|
7409 by a figure similar and similarly situated to that contained by the original straight lines. |
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|
7410 </quote> |
|
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|
7411 </p> |
|
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|
7412 </emph>] |
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|
7413 <lb/><lb/></s></p></div> |
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|
7414 <p xml:lang="lat"> |
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7415 <s xml:id="echoid-s666" xml:space="preserve"> |
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7416 Nota quod in ista <lb/> |
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|
7417 diagrammate præter <lb/> |
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7418 constructionem pappi, <lb/> |
|
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7419 notum etiam Apollonij: <lb/> |
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7420 prop. 54. lib. 1. conicorum. |
|
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7421 <lb/>[<emph style="it">tr: |
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7422 Note that in this diagram besides is the construction of Pappus, |
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7423 noted also by Apollonius in Proposition 54 of Book I on conics. |
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7424 </emph>]<lb/> |
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|
7425 </s> |
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7426 </p> |
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7427 <pb file="add_6787_f569v" o="569v" n="1137"/> |
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|
7428 <pb file="add_6787_f570" o="570" n="1138"/> |
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|
7429 <pb file="add_6787_f570v" o="570v" n="1139"/> |
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7430 <pb file="add_6787_f571" o="571" n="1140"/> |
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|
7431 <pb file="add_6787_f571v" o="571v" n="1141"/> |
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|
7432 <pb file="add_6787_f572" o="572" n="1142"/> |
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7433 <pb file="add_6787_f572v" o="572v" n="1143"/> |
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7434 <pb file="add_6787_f573" o="573" n="1144"/> |
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|
7435 <pb file="add_6787_f573v" o="573v" n="1145"/> |
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7436 <pb file="add_6787_f574" o="574" n="1146"/> |
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7437 <pb file="add_6787_f574v" o="574v" n="1147"/> |
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7438 <pb file="add_6787_f575" o="575" n="1148"/> |
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|
7439 <pb file="add_6787_f575v" o="575v" n="1149"/> |
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7440 <pb file="add_6787_f576" o="576" n="1150"/> |
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|
7441 <pb file="add_6787_f576v" o="576v" n="1151"/> |
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7442 <pb file="add_6787_f577" o="577" n="1152"/> |
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7443 <div xml:id="echoid-div137" type="page_commentary" level="2" n="137"> |
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|
7444 <p> |
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7445 <s xml:id="echoid-s667" xml:space="preserve">[<emph style="it">Note: |
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|
7446 <p> |
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|
7447 <s xml:id="echoid-s667" xml:space="preserve"> |
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7448 The proposition on page 37 of Apollonius, as edited by Commandino in |
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7449 <emph style="it">Conicorum libri quattuor</emph> (1566), is I.52. |
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7450 </s> |
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7451 <lb/> |
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7452 <quote> |
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|
7453 I.52. |
|
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7454 Given a straight line in a plane bounded at one point, to find in the plane the section of a cone |
|
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7455 called parabola, whose diameter is the given straight line, and whose vertex is the end of the straight line, |
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7456 and where whatever straight line is dropped from the section to the diameter at a given angle, |
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|
7457 will equal in square the rectangle contained by the straight line cut off by it from the vertex of the section |
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|
7458 and by some other given straight line. |
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|
7459 </quote> |
|
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|
7460 </p> |
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|
7461 </emph>] |
|
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|
7462 <lb/><lb/></s></p></div> |
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|
7463 <head xml:id="echoid-head129" xml:space="preserve" xml:lang="lat"> |
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|
7464 In Dato Cono: invenire datam parabolam ad pag: 37. Appol. |
|
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|
7465 <lb/>[<emph style="it">tr: |
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|
7466 In a given cone, to find a given parabola, as page 37 of Apollonius. |
|
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|
7467 </emph>]<lb/> |
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|
7468 </head> |
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|
7469 <p xml:lang="lat"> |
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|
7470 <s xml:id="echoid-s669" xml:space="preserve"> |
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|
7471 Dato angulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math>. (Coni) <lb/> |
|
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|
7472 et linea. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. (recta.) <lb/> |
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|
7473 Invenire Isoscelem <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>B</mi><mi>E</mi></mstyle></math> <lb/> |
|
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|
7474 ita ut fit: |
|
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|
7475 <lb/>[<emph style="it">tr: |
|
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|
7476 Given angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math> in the cone and the stright line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, find the isosceles triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>B</mi><mi>E</mi></mstyle></math> such that: |
|
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|
7477 </emph>]<lb/> |
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|
7478 </s> |
|
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|
7479 </p> |
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|
7480 <pb file="add_6787_f577v" o="577v" n="1153"/> |
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|
7481 <pb file="add_6787_f578" o="578" n="1154"/> |
|
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|
7482 <pb file="add_6787_f578v" o="578v" n="1155"/> |
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|
7483 </div> |
|
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|
7484 </text> |
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|
7485 </echo> |