<?xml version="1.0" encoding="utf-8"?>
<contents ref="Perseus:text:1999.01.0086" lang="en">
  <chunk pos="1" start="559385" end="734746" type="book" n="1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1">
    <head lang="en">
      <head>BOOK I.</head>
    </head>
    <chunk pos="1" start="559490" end="565657" type="type" n="Def" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef">
      <head lang="en">
        <head>DEFINITIONS.</head>
      </head>
      <chunk pos="1" start="559585" end="559756" type="number" n="1" id="elem.1.def.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D1">
        <head lang="en">
          <head>1</head>
        </head>
      </chunk>
      <chunk pos="2" start="559762" end="559928" type="number" n="2" id="elem.1.def.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D2">
        <head lang="en">
          <head>2</head>
        </head>
      </chunk>
      <chunk pos="3" start="559934" end="560095" type="number" n="3" id="elem.1.def.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D3">
        <head lang="en">
          <head>3</head>
        </head>
      </chunk>
      <chunk pos="4" start="560101" end="560308" type="number" n="4" id="elem.1.def.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D4">
        <head lang="en">
          <head>4</head>
        </head>
      </chunk>
      <chunk pos="5" start="560314" end="560503" type="number" n="5" id="elem.1.def.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D5">
        <head lang="en">
          <head>5</head>
        </head>
      </chunk>
      <chunk pos="6" start="560509" end="560672" type="number" n="6" id="elem.1.def.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D6">
        <head lang="en">
          <head>6</head>
        </head>
      </chunk>
      <chunk pos="7" start="560678" end="560896" type="number" n="7" id="elem.1.def.7" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D7">
        <head lang="en">
          <head>7</head>
        </head>
      </chunk>
      <chunk pos="8" start="560902" end="561168" type="number" n="8" id="elem.1.def.8" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D8">
        <head lang="en">
          <head>8</head>
        </head>
      </chunk>
      <chunk pos="9" start="561174" end="561397" type="number" n="9" id="elem.1.def.9" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D9">
        <head lang="en">
          <head>9</head>
        </head>
      </chunk>
      <chunk pos="10" start="561403" end="561783" type="number" n="10" id="elem.1.def.10" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D10">
        <head lang="en">
          <head>10</head>
        </head>
      </chunk>
      <chunk pos="11" start="561789" end="561984" type="number" n="11" id="elem.1.def.11" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D11">
        <head lang="en">
          <head>11</head>
        </head>
      </chunk>
      <chunk pos="12" start="561990" end="562181" type="number" n="12" id="elem.1.def.12" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D12">
        <head lang="en">
          <head>12</head>
        </head>
      </chunk>
      <chunk pos="13" start="562187" end="562380" type="number" n="13" id="elem.1.def.13" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D13">
        <head lang="en">
          <head>13</head>
        </head>
      </chunk>
      <chunk pos="14" start="562386" end="562592" type="number" n="14" id="elem.1.def.14" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D14">
        <head lang="en">
          <head>14</head>
        </head>
      </chunk>
      <chunk pos="15" start="562598" end="562912" type="number" n="15" id="elem.1.def.15" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D15">
        <head lang="en">
          <head>15</head>
        </head>
      </chunk>
      <chunk pos="16" start="562918" end="563128" type="number" n="16" id="elem.1.def.16" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D16">
        <head lang="en">
          <head>16</head>
        </head>
      </chunk>
      <chunk pos="17" start="563134" end="563464" type="number" n="17" id="elem.1.def.17" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D17">
        <head lang="en">
          <head>17</head>
        </head>
      </chunk>
      <chunk pos="18" start="563470" end="563767" type="number" n="18" id="elem.1.def.18" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D18">
        <head lang="en">
          <head>18</head>
        </head>
      </chunk>
      <chunk pos="19" start="563773" end="564125" type="number" n="19" id="elem.1.def.19" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D19">
        <head lang="en">
          <head>19</head>
        </head>
      </chunk>
      <chunk pos="20" start="564131" end="564475" type="number" n="20" id="elem.1.def.20" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D20">
        <head lang="en">
          <head>20</head>
        </head>
      </chunk>
      <chunk pos="21" start="564481" end="564821" type="number" n="21" id="elem.1.def.21" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D21">
        <head lang="en">
          <head>21</head>
        </head>
      </chunk>
      <chunk pos="22" start="564827" end="565342" type="number" n="22" id="elem.1.def.22" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D22">
        <head lang="en">
          <head>22</head>
        </head>
      </chunk>
      <chunk pos="23" start="565348" end="565645" type="number" n="23" id="elem.1.def.23" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D23">
        <head lang="en">
          <head>23</head>
        </head>
      </chunk>
    </chunk>
    <chunk pos="2" start="565662" end="566948" type="type" n="Post" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DPost">
      <head lang="en">
        <head>POSTULATES.</head>
      </head>
      <chunk pos="24" start="565757" end="565980" type="number" n="1" id="elem.1.post.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DPost%3Anumber%3D1">
        <head lang="en">
          <head>1</head>
        </head>
      </chunk>
      <chunk pos="25" start="565986" end="566177" type="number" n="2" id="elem.1.post.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DPost%3Anumber%3D2">
        <head lang="en">
          <head>2</head>
        </head>
      </chunk>
      <chunk pos="26" start="566183" end="566358" type="number" n="3" id="elem.1.post.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DPost%3Anumber%3D3">
        <head lang="en">
          <head>3</head>
        </head>
      </chunk>
      <chunk pos="27" start="566364" end="566536" type="number" n="4" id="elem.1.post.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DPost%3Anumber%3D4">
        <head lang="en">
          <head>4</head>
        </head>
      </chunk>
      <chunk pos="28" start="566542" end="566936" type="number" n="5" id="elem.1.post.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DPost%3Anumber%3D5">
        <head lang="en">
          <head>5</head>
        </head>
      </chunk>
    </chunk>
    <chunk pos="3" start="566953" end="567997" type="type" n="CN" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DCN">
      <head lang="en">
        <head>COMMON NOTIONS.</head>
      </head>
      <chunk pos="29" start="567050" end="567245" type="number" n="1" id="elem.1.c.n.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DCN%3Anumber%3D1">
        <head lang="en">
          <head>1</head>
        </head>
      </chunk>
      <chunk pos="30" start="567251" end="567426" type="number" n="2" id="elem.1.c.n.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DCN%3Anumber%3D2">
        <head lang="en">
          <head>2</head>
        </head>
      </chunk>
      <chunk pos="31" start="567432" end="567618" type="number" n="3" id="elem.1.c.n.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DCN%3Anumber%3D3">
        <head lang="en">
          <head>3</head>
        </head>
      </chunk>
      <chunk pos="32" start="567624" end="567816" type="number" n="4" id="elem.1.c.n.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DCN%3Anumber%3D4">
        <head lang="en">
          <head>4</head>
        </head>
      </chunk>
      <chunk pos="33" start="567822" end="567985" type="number" n="5" id="elem.1.c.n.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DCN%3Anumber%3D5">
        <head lang="en">
          <head>5</head>
        </head>
      </chunk>
    </chunk>
    <chunk pos="4" start="568002" end="734735" type="type" n="Prop" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp">
      <head lang="en">
        <head>BOOK I. PROPOSITIONS.</head>
      </head>
      <chunk pos="34" start="568127" end="574168" type="number" n="1" id="elem.1.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D1">
        <head lang="en">
          <head>Proposition 1.</head>
        </head>
      </chunk>
      <chunk pos="35" start="574174" end="578864" type="number" n="2" id="elem.1.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D2">
        <head lang="en">
          <head>Proposition 2.</head>
        </head>
      </chunk>
      <chunk pos="36" start="578870" end="580644" type="number" n="3" id="elem.1.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D3">
        <head lang="en">
          <head>Proposition 3.</head>
        </head>
      </chunk>
      <chunk pos="37" start="580650" end="590690" type="number" n="4" id="elem.1.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D4">
        <head lang="en">
          <head>Proposition 4.</head>
        </head>
      </chunk>
      <chunk pos="38" start="590696" end="597234" type="number" n="5" id="elem.1.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D5">
        <head lang="en">
          <head>Proposition 5.</head>
        </head>
      </chunk>
      <chunk pos="39" start="597240" end="598928" type="number" n="6" id="elem.1.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D6">
        <head lang="en">
          <head>Proposition 6.</head>
        </head>
      </chunk>
      <chunk pos="40" start="598934" end="607273" type="number" n="7" id="elem.1.7" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D7">
        <head lang="en">
          <head>Proposition 7.</head>
        </head>
      </chunk>
      <chunk pos="41" start="607279" end="610591" type="number" n="8" id="elem.1.8" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D8">
        <head lang="en">
          <head>Proposition 8.</head>
        </head>
      </chunk>
      <chunk pos="42" start="610597" end="612204" type="number" n="9" id="elem.1.9" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D9">
        <head lang="en">
          <head>Proposition 9.</head>
        </head>
      </chunk>
      <chunk pos="43" start="612210" end="613740" type="number" n="10" id="elem.1.10" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D10">
        <head lang="en">
          <head>Proposition 10.</head>
        </head>
      </chunk>
      <chunk pos="44" start="613746" end="616566" type="number" n="11" id="elem.1.11" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D11">
        <head lang="en">
          <head>Proposition 11.</head>
        </head>
      </chunk>
      <chunk pos="45" start="616572" end="620372" type="number" n="12" id="elem.1.12" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D12">
        <head lang="en">
          <head>Proposition 12.</head>
        </head>
      </chunk>
      <chunk pos="46" start="620378" end="624109" type="number" n="13" id="elem.1.13" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D13">
        <head lang="en">
          <head>Proposition 13.</head>
        </head>
      </chunk>
      <chunk pos="47" start="624115" end="628753" type="number" n="14" id="elem.1.14" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D14">
        <head lang="en">
          <head>Proposition 14.</head>
        </head>
      </chunk>
      <chunk pos="48" start="628759" end="632476" type="number" n="15" id="elem.1.15" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D15">
        <head lang="en">
          <head>Proposition 15.</head>
        </head>
      </chunk>
      <chunk pos="49" start="632482" end="636039" type="number" n="16" id="elem.1.16" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D16">
        <head lang="en">
          <head>Proposition 16.</head>
        </head>
      </chunk>
      <chunk pos="50" start="636045" end="637927" type="number" n="17" id="elem.1.17" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D17">
        <head lang="en">
          <head>Proposition 17.</head>
        </head>
      </chunk>
      <chunk pos="51" start="637933" end="641080" type="number" n="18" id="elem.1.18" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D18">
        <head lang="en">
          <head>Proposition 18.</head>
        </head>
      </chunk>
      <chunk pos="52" start="641086" end="642647" type="number" n="19" id="elem.1.19" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D19">
        <head lang="en">
          <head>Proposition 19.</head>
        </head>
      </chunk>
      <chunk pos="53" start="642653" end="645822" type="number" n="20" id="elem.1.20" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D20">
        <head lang="en">
          <head>Proposition 20.</head>
        </head>
      </chunk>
      <chunk pos="54" start="645828" end="649587" type="number" n="21" id="elem.1.21" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D21">
        <head lang="en">
          <head>Proposition 21.</head>
        </head>
      </chunk>
      <chunk pos="55" start="649593" end="654460" type="number" n="22" id="elem.1.22" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D22">
        <head lang="en">
          <head>Proposition 22.</head>
        </head>
      </chunk>
      <chunk pos="56" start="654466" end="656416" type="number" n="23" id="elem.1.23" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D23">
        <head lang="en">
          <head>Proposition 23.</head>
        </head>
      </chunk>
      <chunk pos="57" start="656422" end="660289" type="number" n="24" id="elem.1.24" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D24">
        <head lang="en">
          <head>Proposition 24.</head>
        </head>
      </chunk>
      <chunk pos="58" start="660295" end="662308" type="number" n="25" id="elem.1.25" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D25">
        <head lang="en">
          <head>Proposition 25.</head>
        </head>
      </chunk>
      <chunk pos="59" start="662314" end="668852" type="number" n="26" id="elem.1.26" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D26">
        <head lang="en">
          <head>Proposition 26.</head>
        </head>
      </chunk>
      <chunk pos="60" start="668858" end="672539" type="number" n="27" id="elem.1.27" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D27">
        <head lang="en">
          <head>Proposition 27.</head>
        </head>
      </chunk>
      <chunk pos="61" start="672545" end="674749" type="number" n="28" id="elem.1.28" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D28">
        <head lang="en">
          <head>Proposition 28.</head>
        </head>
      </chunk>
      <chunk pos="62" start="674755" end="678746" type="number" n="29" id="elem.1.29" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D29">
        <head lang="en">
          <head>Proposition 29.</head>
        </head>
      </chunk>
      <chunk pos="63" start="678752" end="680707" type="number" n="30" id="elem.1.30" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D30">
        <head lang="en">
          <head>Proposition 30.</head>
        </head>
      </chunk>
      <chunk pos="64" start="680713" end="682351" type="number" n="31" id="elem.1.31" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D31">
        <head lang="en">
          <head>Proposition 31.</head>
        </head>
      </chunk>
      <chunk pos="65" start="682357" end="684900" type="number" n="32" id="elem.1.32" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D32">
        <head lang="en">
          <head>Proposition 32.</head>
        </head>
      </chunk>
      <chunk pos="66" start="684906" end="688551" type="number" n="33" id="elem.1.33" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D33">
        <head lang="en">
          <head>Proposition 33.</head>
        </head>
      </chunk>
      <chunk pos="67" start="688557" end="693439" type="number" n="34" id="elem.1.34" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D34">
        <head lang="en">
          <head>Proposition 34.</head>
        </head>
      </chunk>
      <chunk pos="68" start="693445" end="696555" type="number" n="35" id="elem.1.35" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D35">
        <head lang="en">
          <head>Proposition 35.</head>
        </head>
      </chunk>
      <chunk pos="69" start="696561" end="698501" type="number" n="36" id="elem.1.36" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D36">
        <head lang="en">
          <head>Proposition 36.</head>
        </head>
      </chunk>
      <chunk pos="70" start="698507" end="701066" type="number" n="37" id="elem.1.37" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D37">
        <head lang="en">
          <head>Proposition 37.</head>
        </head>
      </chunk>
      <chunk pos="71" start="701072" end="703064" type="number" n="38" id="elem.1.38" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D38">
        <head lang="en">
          <head>Proposition 38.</head>
        </head>
      </chunk>
      <chunk pos="72" start="703070" end="705830" type="number" n="39" id="elem.1.39" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D39">
        <head lang="en">
          <head>Proposition 39.</head>
        </head>
      </chunk>
      <chunk pos="73" start="705836" end="707468" type="number" n="40" id="elem.1.40" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D40">
        <head lang="en">
          <head>[Proposition 40.</head>
        </head>
      </chunk>
      <chunk pos="74" start="707474" end="708935" type="number" n="41" id="elem.1.41" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D41">
        <head lang="en">
          <head>Proposition 41.</head>
        </head>
      </chunk>
      <chunk pos="75" start="708941" end="711221" type="number" n="42" id="elem.1.42" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D42">
        <head lang="en">
          <head>Proposition 42.</head>
        </head>
      </chunk>
      <chunk pos="76" start="711227" end="714222" type="number" n="43" id="elem.1.43" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D43">
        <head lang="en">
          <head>Proposition 43.</head>
        </head>
      </chunk>
      <chunk pos="77" start="714228" end="718015" type="number" n="44" id="elem.1.44" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D44">
        <head lang="en">
          <head>Proposition 44.</head>
        </head>
      </chunk>
      <chunk pos="78" start="718021" end="722907" type="number" n="45" id="elem.1.45" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D45">
        <head lang="en">
          <head>Proposition 45.</head>
        </head>
      </chunk>
      <chunk pos="79" start="722913" end="726360" type="number" n="46" id="elem.1.46" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D46">
        <head lang="en">
          <head>Proposition 46.</head>
        </head>
      </chunk>
      <chunk pos="80" start="726366" end="732239" type="number" n="47" id="elem.1.47" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D47">
        <head lang="en">
          <head>Proposition 47.</head>
        </head>
      </chunk>
      <chunk pos="81" start="732245" end="734723" type="number" n="48" id="elem.1.48" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D48">
        <head lang="en">
          <head>Proposition 48.</head>
        </head>
      </chunk>
    </chunk>
  </chunk>
  <chunk pos="2" start="734750" end="784870" type="book" n="2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D2">
    <head lang="en">
      <head>BOOK II.</head>
    </head>
    <chunk pos="5" start="734856" end="735505" type="type" n="Def" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D2%3Atype%3DDef">
      <head lang="en">
        <head>DEFINITIONS.</head>
      </head>
      <chunk pos="82" start="734951" end="735203" type="number" n="1" id="elem.2.def.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D2%3Atype%3DDef%3Anumber%3D1">
        <head lang="en">
          <head>1</head>
        </head>
      </chunk>
      <chunk pos="83" start="735209" end="735493" type="number" n="2" id="elem.2.def.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D2%3Atype%3DDef%3Anumber%3D2">
        <head lang="en">
          <head>2</head>
        </head>
      </chunk>
    </chunk>
    <chunk pos="6" start="735510" end="784859" type="type" n="Prop" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D2%3Atype%3DProp">
      <head lang="en">
        <head>BOOK II. PROPOSITIONS.</head>
      </head>
    </chunk>
  </chunk>
  <chunk pos="3" start="784911" end="891926" type="book" n="3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3">
    <head lang="en">
      <head>BOOK III.</head>
    </head>
    <chunk pos="7" start="785016" end="788131" type="type" n="Def" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DDef">
      <head lang="en">
        <head>DEFINITIONS.</head>
      </head>
      <chunk pos="84" start="785111" end="785346" type="number" n="1" id="elem.3.def.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DDef%3Anumber%3D1">
        <head lang="en">
          <head>1</head>
        </head>
      </chunk>
      <chunk pos="85" start="785352" end="785604" type="number" n="2" id="elem.3.def.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DDef%3Anumber%3D2">
        <head lang="en">
          <head>2</head>
        </head>
      </chunk>
      <chunk pos="86" start="785610" end="785860" type="number" n="3" id="elem.3.def.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DDef%3Anumber%3D3">
        <head lang="en">
          <head>3</head>
        </head>
      </chunk>
      <chunk pos="87" start="785866" end="786144" type="number" n="4" id="elem.3.def.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DDef%3Anumber%3D4">
        <head lang="en">
          <head>4</head>
        </head>
      </chunk>
      <chunk pos="88" start="786150" end="786383" type="number" n="5" id="elem.3.def.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DDef%3Anumber%3D5">
        <head lang="en">
          <head>5</head>
        </head>
      </chunk>
      <chunk pos="89" start="786389" end="786619" type="number" n="6" id="elem.3.def.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DDef%3Anumber%3D6">
        <head lang="en">
          <head>6</head>
        </head>
      </chunk>
      <chunk pos="90" start="786625" end="786849" type="number" n="7" id="elem.3.def.7" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DDef%3Anumber%3D7">
        <head lang="en">
          <head>7</head>
        </head>
      </chunk>
      <chunk pos="91" start="786855" end="787266" type="number" n="8" id="elem.3.def.8" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DDef%3Anumber%3D8">
        <head lang="en">
          <head>8</head>
        </head>
      </chunk>
      <chunk pos="92" start="787272" end="787531" type="number" n="9" id="elem.3.def.9" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DDef%3Anumber%3D9">
        <head lang="en">
          <head>9</head>
        </head>
      </chunk>
      <chunk pos="93" start="787537" end="787865" type="number" n="10" id="elem.3.def.10" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DDef%3Anumber%3D10">
        <head lang="en">
          <head>10</head>
        </head>
      </chunk>
      <chunk pos="94" start="787871" end="788119" type="number" n="11" id="elem.3.def.11" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DDef%3Anumber%3D11">
        <head lang="en">
          <head>11</head>
        </head>
      </chunk>
    </chunk>
    <chunk pos="8" start="788136" end="891915" type="type" n="Prop" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp">
      <head lang="en">
        <head>BOOK III. PROPOSITIONS.</head>
      </head>
      <chunk pos="95" start="788261" end="791439" type="number" n="1" id="elem.3.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D1">
        <head lang="en">
          <head>PROPOSITION 1.</head>
        </head>
      </chunk>
      <chunk pos="96" start="791445" end="793699" type="number" n="2" id="elem.3.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D2">
        <head lang="en">
          <head>PROPOSITION 2.</head>
        </head>
      </chunk>
      <chunk pos="97" start="793705" end="796616" type="number" n="3" id="elem.3.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D3">
        <head lang="en">
          <head>PROPOSITION 3.</head>
        </head>
      </chunk>
      <chunk pos="98" start="796622" end="798382" type="number" n="4" id="elem.3.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D4">
        <head lang="en">
          <head>PROPOSITION 4.</head>
        </head>
      </chunk>
      <chunk pos="99" start="798388" end="799723" type="number" n="5" id="elem.3.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D5">
        <head lang="en">
          <head>PROPOSITION 5.</head>
        </head>
      </chunk>
      <chunk pos="100" start="799729" end="800998" type="number" n="6" id="elem.3.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D6">
        <head lang="en">
          <head>PROPOSITION 6.</head>
        </head>
      </chunk>
      <chunk pos="101" start="801004" end="806417" type="number" n="7" id="elem.3.7" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D7">
        <head lang="en">
          <head>PROPOSITION 7.</head>
        </head>
      </chunk>
      <chunk pos="102" start="806423" end="812468" type="number" n="8" id="elem.3.8" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D8">
        <head lang="en">
          <head>PROPOSITION 8.</head>
        </head>
      </chunk>
      <chunk pos="103" start="812474" end="814846" type="number" n="9" id="elem.3.9" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D9">
        <head lang="en">
          <head>PROPOSITION 9.</head>
        </head>
      </chunk>
      <chunk pos="104" start="814852" end="817192" type="number" n="10" id="elem.3.10" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D10">
        <head lang="en">
          <head>PROPOSITION 10.</head>
        </head>
      </chunk>
      <chunk pos="105" start="817198" end="819430" type="number" n="11" id="elem.3.11" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D11">
        <head lang="en">
          <head>PROPOSITION 11.</head>
        </head>
      </chunk>
      <chunk pos="106" start="819436" end="821637" type="number" n="12" id="elem.3.12" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D12">
        <head lang="en">
          <head>PROPOSITION 12.</head>
        </head>
      </chunk>
      <chunk pos="107" start="821643" end="824846" type="number" n="13" id="elem.3.13" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D13">
        <head lang="en">
          <head>PROPOSITION 13.</head>
        </head>
      </chunk>
      <chunk pos="108" start="824852" end="829099" type="number" n="14" id="elem.3.14" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D14">
        <head lang="en">
          <head>PROPOSITION 14.</head>
        </head>
      </chunk>
      <chunk pos="109" start="829105" end="832082" type="number" n="15" id="elem.3.15" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D15">
        <head lang="en">
          <head>PROPOSITION 15.</head>
        </head>
      </chunk>
      <chunk pos="110" start="832088" end="836904" type="number" n="16" id="elem.3.16" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D16">
        <head lang="en">
          <head>PROPOSITION 16.</head>
        </head>
      </chunk>
      <chunk pos="111" start="836910" end="839197" type="number" n="17" id="elem.3.17" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D17">
        <head lang="en">
          <head>PROPOSITION 17.</head>
        </head>
      </chunk>
      <chunk pos="112" start="839203" end="841149" type="number" n="18" id="elem.3.18" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D18">
        <head lang="en">
          <head>PROPOSITION 18.</head>
        </head>
      </chunk>
      <chunk pos="113" start="841155" end="842707" type="number" n="19" id="elem.3.19" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D19">
        <head lang="en">
          <head>PROPOSITION 19.</head>
        </head>
      </chunk>
      <chunk pos="114" start="842713" end="845733" type="number" n="20" id="elem.3.20" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D20">
        <head lang="en">
          <head>PROPOSITION 20.</head>
        </head>
      </chunk>
      <chunk pos="115" start="845739" end="846960" type="number" n="21" id="elem.3.21" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D21">
        <head lang="en">
          <head>PROPOSITION 21.</head>
        </head>
      </chunk>
      <chunk pos="116" start="846966" end="848721" type="number" n="22" id="elem.3.22" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D22">
        <head lang="en">
          <head>PROPOSITION 22.</head>
        </head>
      </chunk>
      <chunk pos="117" start="848727" end="849976" type="number" n="23" id="elem.3.23" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D23">
        <head lang="en">
          <head>PROPOSITION 23.</head>
        </head>
      </chunk>
      <chunk pos="118" start="849982" end="852090" type="number" n="24" id="elem.3.24" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D24">
        <head lang="en">
          <head>PROPOSITION 24.</head>
        </head>
      </chunk>
      <chunk pos="119" start="852096" end="855902" type="number" n="25" id="elem.3.25" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D25">
        <head lang="en">
          <head>PROPOSITION 25.</head>
        </head>
      </chunk>
      <chunk pos="120" start="855908" end="857905" type="number" n="26" id="elem.3.26" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D26">
        <head lang="en">
          <head>PROPOSITION 26.</head>
        </head>
      </chunk>
      <chunk pos="121" start="857911" end="860070" type="number" n="27" id="elem.3.27" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D27">
        <head lang="en">
          <head>PROPOSITION 27.</head>
        </head>
      </chunk>
      <chunk pos="122" start="860076" end="861976" type="number" n="28" id="elem.3.28" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D28">
        <head lang="en">
          <head>PROPOSITION 28.</head>
        </head>
      </chunk>
      <chunk pos="123" start="861982" end="863397" type="number" n="29" id="elem.3.29" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D29">
        <head lang="en">
          <head>PROPOSITION 29.</head>
        </head>
      </chunk>
      <chunk pos="124" start="863403" end="864938" type="number" n="30" id="elem.3.30" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D30">
        <head lang="en">
          <head>PROPOSITION 30.</head>
        </head>
      </chunk>
      <chunk pos="125" start="864944" end="868927" type="number" n="31" id="elem.3.31" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D31">
        <head lang="en">
          <head>PROPOSITION 31.</head>
        </head>
      </chunk>
      <chunk pos="126" start="868933" end="872068" type="number" n="32" id="elem.3.32" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D32">
        <head lang="en">
          <head>PROPOSITION 32.</head>
        </head>
      </chunk>
      <chunk pos="127" start="872074" end="878511" type="number" n="33" id="elem.3.33" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D33">
        <head lang="en">
          <head>PROPOSITION 33.</head>
        </head>
      </chunk>
      <chunk pos="128" start="878517" end="880159" type="number" n="34" id="elem.3.34" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D34">
        <head lang="en">
          <head>PROPOSITION 34.</head>
        </head>
      </chunk>
      <chunk pos="129" start="880165" end="883928" type="number" n="35" id="elem.3.35" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D35">
        <head lang="en">
          <head>PROPOSITION 35.</head>
        </head>
      </chunk>
      <chunk pos="130" start="883934" end="888890" type="number" n="36" id="elem.3.36" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D36">
        <head lang="en">
          <head>PROPOSITION 36.</head>
        </head>
      </chunk>
      <chunk pos="131" start="888896" end="891903" type="number" n="37" id="elem.3.37" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D3%3Atype%3DProp%3Anumber%3D37">
        <head lang="en">
          <head>PROPOSITION 37.</head>
        </head>
      </chunk>
    </chunk>
  </chunk>
  <chunk pos="4" start="891930" end="948563" type="book" n="4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4">
    <head lang="en">
      <head>BOOK IV.</head>
    </head>
    <chunk pos="9" start="892035" end="894187" type="type" n="Def" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DDef">
      <head lang="en">
        <head>DEFINITIONS.</head>
      </head>
      <chunk pos="132" start="892130" end="892444" type="number" n="1" id="elem.4.def.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DDef%3Anumber%3D1">
        <head lang="en">
          <head>1</head>
        </head>
      </chunk>
      <chunk pos="133" start="892450" end="892774" type="number" n="2" id="elem.4.def.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DDef%3Anumber%3D2">
        <head lang="en">
          <head>2</head>
        </head>
      </chunk>
      <chunk pos="134" start="892780" end="893050" type="number" n="3" id="elem.4.def.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DDef%3Anumber%3D3">
        <head lang="en">
          <head>3</head>
        </head>
      </chunk>
      <chunk pos="135" start="893056" end="893337" type="number" n="4" id="elem.4.def.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DDef%3Anumber%3D4">
        <head lang="en">
          <head>4</head>
        </head>
      </chunk>
      <chunk pos="136" start="893343" end="893625" type="number" n="5" id="elem.4.def.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DDef%3Anumber%3D5">
        <head lang="en">
          <head>5</head>
        </head>
      </chunk>
      <chunk pos="137" start="893631" end="893925" type="number" n="6" id="elem.4.def.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DDef%3Anumber%3D6">
        <head lang="en">
          <head>6</head>
        </head>
      </chunk>
      <chunk pos="138" start="893931" end="894175" type="number" n="7" id="elem.4.def.7" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DDef%3Anumber%3D7">
        <head lang="en">
          <head>7</head>
        </head>
      </chunk>
    </chunk>
    <chunk pos="10" start="894192" end="948552" type="type" n="Prop" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DProp">
      <head lang="en">
        <head>BOOK IV. PROPOSITIONS</head>
      </head>
      <chunk pos="139" start="894316" end="895910" type="number" n="1" id="elem.4.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DProp%3Anumber%3D1">
        <head lang="en">
          <head>PROPOSITION 1.</head>
        </head>
      </chunk>
      <chunk pos="140" start="895916" end="897910" type="number" n="2" id="elem.4.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DProp%3Anumber%3D2">
        <head lang="en">
          <head>PROPOSITION 2.</head>
        </head>
      </chunk>
      <chunk pos="141" start="897916" end="901788" type="number" n="3" id="elem.4.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DProp%3Anumber%3D3">
        <head lang="en">
          <head>PROPOSITION 3.</head>
        </head>
      </chunk>
      <chunk pos="142" start="901794" end="905817" type="number" n="4" id="elem.4.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DProp%3Anumber%3D4">
        <head lang="en">
          <head>PROPOSITION 4.</head>
        </head>
      </chunk>
      <chunk pos="143" start="905823" end="909585" type="number" n="5" id="elem.4.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DProp%3Anumber%3D5">
        <head lang="en">
          <head>PROPOSITION 5.</head>
        </head>
      </chunk>
      <chunk pos="144" start="909591" end="911556" type="number" n="6" id="elem.4.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DProp%3Anumber%3D6">
        <head lang="en">
          <head>PROPOSITION 6.</head>
        </head>
      </chunk>
      <chunk pos="145" start="911562" end="914849" type="number" n="7" id="elem.4.7" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DProp%3Anumber%3D7">
        <head lang="en">
          <head>PROPOSITION 7.</head>
        </head>
      </chunk>
      <chunk pos="146" start="914855" end="917860" type="number" n="8" id="elem.4.8" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DProp%3Anumber%3D8">
        <head lang="en">
          <head>PROPOSITION 8.</head>
        </head>
      </chunk>
      <chunk pos="147" start="917866" end="920218" type="number" n="9" id="elem.4.9" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DProp%3Anumber%3D9">
        <head lang="en">
          <head>PROPOSITION 9.</head>
        </head>
      </chunk>
      <chunk pos="148" start="920224" end="924555" type="number" n="10" id="elem.4.10" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DProp%3Anumber%3D10">
        <head lang="en">
          <head>PROPOSITION 10.</head>
        </head>
      </chunk>
      <chunk pos="149" start="924561" end="928032" type="number" n="11" id="elem.4.11" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DProp%3Anumber%3D11">
        <head lang="en">
          <head>PROPOSITION 11.</head>
        </head>
      </chunk>
      <chunk pos="150" start="928038" end="934160" type="number" n="12" id="elem.4.12" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DProp%3Anumber%3D12">
        <head lang="en">
          <head>PROPOSITION 12.</head>
        </head>
      </chunk>
      <chunk pos="151" start="934166" end="938898" type="number" n="13" id="elem.4.13" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DProp%3Anumber%3D13">
        <head lang="en">
          <head>PROPOSITION 13.</head>
        </head>
      </chunk>
      <chunk pos="152" start="938904" end="941276" type="number" n="14" id="elem.4.14" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DProp%3Anumber%3D14">
        <head lang="en">
          <head>PROPOSITION 14.</head>
        </head>
      </chunk>
      <chunk pos="153" start="941282" end="946526" type="number" n="15" id="elem.4.15" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DProp%3Anumber%3D15">
        <head lang="en">
          <head>PROPOSITION 15.</head>
        </head>
      </chunk>
      <chunk pos="154" start="946532" end="948540" type="number" n="16" id="elem.4.16" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D4%3Atype%3DProp%3Anumber%3D16">
        <head lang="en">
          <head>PROPOSITION 16.</head>
        </head>
      </chunk>
    </chunk>
  </chunk>
  <chunk pos="5" start="948567" end="1024553" type="book" n="5" id="elem.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5">
    <head lang="en">
      <head>BOOK V.</head>
    </head>
    <chunk pos="11" start="948684" end="954285" type="type" n="Def" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DDef">
      <head lang="en">
        <head>DEFINITIONS.</head>
      </head>
      <chunk pos="155" start="948779" end="949004" type="number" n="1" id="elem.5.def.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DDef%3Anumber%3D1">
        <head lang="en">
          <head>1</head>
        </head>
      </chunk>
      <chunk pos="156" start="949010" end="949213" type="number" n="2" id="elem.5.def.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DDef%3Anumber%3D2">
        <head lang="en">
          <head>2</head>
        </head>
      </chunk>
      <chunk pos="157" start="949219" end="949462" type="number" n="3" id="elem.5.def.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DDef%3Anumber%3D3">
        <head lang="en">
          <head>3</head>
        </head>
      </chunk>
      <chunk pos="158" start="949468" end="949713" type="number" n="4" id="elem.5.def.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DDef%3Anumber%3D4">
        <head lang="en">
          <head>4</head>
        </head>
      </chunk>
      <chunk pos="159" start="949719" end="950226" type="number" n="5" id="elem.5.def.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DDef%3Anumber%3D5">
        <head lang="en">
          <head>5</head>
        </head>
      </chunk>
      <chunk pos="160" start="950232" end="950429" type="number" n="6" id="elem.5.def.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DDef%3Anumber%3D6">
        <head lang="en">
          <head>6</head>
        </head>
      </chunk>
      <chunk pos="161" start="950435" end="950836" type="number" n="7" id="elem.5.def.7" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DDef%3Anumber%3D7">
        <head lang="en">
          <head>7</head>
        </head>
      </chunk>
      <chunk pos="162" start="950842" end="951004" type="number" n="8" id="elem.5.def.8" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DDef%3Anumber%3D8">
        <head lang="en">
          <head>8</head>
        </head>
      </chunk>
      <chunk pos="163" start="951010" end="951277" type="number" n="9" id="elem.5.def.9" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DDef%3Anumber%3D9">
        <head lang="en">
          <head>9</head>
        </head>
      </chunk>
      <chunk pos="164" start="951283" end="951626" type="number" n="10" id="elem.5.def.10" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DDef%3Anumber%3D10">
        <head lang="en">
          <head>10</head>
        </head>
      </chunk>
      <chunk pos="165" start="951632" end="951899" type="number" n="11" id="elem.5.def.11" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DDef%3Anumber%3D11">
        <head lang="en">
          <head>11</head>
        </head>
      </chunk>
      <chunk pos="166" start="951905" end="952164" type="number" n="12" id="elem.5.def.12" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DDef%3Anumber%3D12">
        <head lang="en">
          <head>12</head>
        </head>
      </chunk>
      <chunk pos="167" start="952170" end="952406" type="number" n="13" id="elem.5.def.13" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DDef%3Anumber%3D13">
        <head lang="en">
          <head>13</head>
        </head>
      </chunk>
      <chunk pos="168" start="952412" end="952696" type="number" n="14" id="elem.5.def.14" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DDef%3Anumber%3D14">
        <head lang="en">
          <head>14</head>
        </head>
      </chunk>
      <chunk pos="169" start="952702" end="952971" type="number" n="15" id="elem.5.def.15" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DDef%3Anumber%3D15">
        <head lang="en">
          <head>15</head>
        </head>
      </chunk>
      <chunk pos="170" start="952977" end="953236" type="number" n="16" id="elem.5.def.16" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DDef%3Anumber%3D16">
        <head lang="en">
          <head>16</head>
        </head>
      </chunk>
      <chunk pos="171" start="953242" end="953776" type="number" n="17" id="elem.5.def.17" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DDef%3Anumber%3D17">
        <head lang="en">
          <head>17</head>
        </head>
      </chunk>
      <chunk pos="172" start="953782" end="954273" type="number" n="18" id="elem.5.def.18" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DDef%3Anumber%3D18">
        <head lang="en">
          <head>18</head>
        </head>
      </chunk>
    </chunk>
    <chunk pos="12" start="954290" end="1024542" type="type" n="Prop" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp">
      <head lang="en">
        <head>BOOK V. PROPOSITIONS.</head>
      </head>
      <chunk pos="173" start="954415" end="956632" type="number" n="1" id="elem.5.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D1">
        <head lang="en">
          <head>PROPOSITION I.</head>
        </head>
      </chunk>
      <chunk pos="174" start="956638" end="958656" type="number" n="2" id="elem.5.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D2">
        <head lang="en">
          <head>PROPOSITION 2.</head>
        </head>
      </chunk>
      <chunk pos="175" start="958662" end="961240" type="number" n="3" id="elem.5.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D3">
        <head lang="en">
          <head>PROPOSITION 3.</head>
        </head>
      </chunk>
      <chunk pos="176" start="961246" end="963798" type="number" n="4" id="elem.5.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D4">
        <head lang="en">
          <head>PROPOSITION 4.</head>
        </head>
      </chunk>
      <chunk pos="177" start="963804" end="966860" type="number" n="5" id="elem.5.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D5">
        <head lang="en">
          <head>PROPOSITION 5.</head>
        </head>
      </chunk>
      <chunk pos="178" start="966866" end="969262" type="number" n="6" id="elem.5.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D6">
        <head lang="en">
          <head>PROPOSITION 6.</head>
        </head>
      </chunk>
      <chunk pos="179" start="969268" end="971948" type="number" n="7" id="elem.5.7" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D7">
        <head lang="en">
          <head>PROPOSITION 7.</head>
        </head>
      </chunk>
      <chunk pos="180" start="971954" end="978189" type="number" n="8" id="elem.5.8" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D8">
        <head lang="en">
          <head>PROPOSITION 8.</head>
        </head>
      </chunk>
      <chunk pos="181" start="978195" end="979463" type="number" n="9" id="elem.5.9" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D9">
        <head lang="en">
          <head>PROPOSITION 9.</head>
        </head>
      </chunk>
      <chunk pos="182" start="979469" end="981996" type="number" n="10" id="elem.5.10" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D10">
        <head lang="en">
          <head>PROPOSITION 10.</head>
        </head>
      </chunk>
      <chunk pos="183" start="982002" end="984434" type="number" n="11" id="elem.5.11" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D11">
        <head lang="en">
          <head>PROPOSITION 11.</head>
        </head>
      </chunk>
      <chunk pos="184" start="984440" end="987232" type="number" n="12" id="elem.5.12" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D12">
        <head lang="en">
          <head>PROPOSITION 12.</head>
        </head>
      </chunk>
      <chunk pos="185" start="987238" end="990220" type="number" n="13" id="elem.5.13" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D13">
        <head lang="en">
          <head>PROPOSITION 13.</head>
        </head>
      </chunk>
      <chunk pos="186" start="990226" end="992069" type="number" n="14" id="elem.5.14" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D14">
        <head lang="en">
          <head>PROPOSITION 14.</head>
        </head>
      </chunk>
      <chunk pos="187" start="992075" end="994206" type="number" n="15" id="elem.5.15" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D15">
        <head lang="en">
          <head>PROPOSITION 15.</head>
        </head>
      </chunk>
      <chunk pos="188" start="994212" end="998621" type="number" n="16" id="elem.5.16" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D16">
        <head lang="en">
          <head>PROPOSITION 16.</head>
        </head>
      </chunk>
      <chunk pos="189" start="998627" end="1002948" type="number" n="17" id="elem.5.17" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D17">
        <head lang="en">
          <head>PROPOSITION 17.</head>
        </head>
      </chunk>
      <chunk pos="190" start="1002954" end="1005470" type="number" n="18" id="elem.5.18" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D18">
        <head lang="en">
          <head>PROPOSITION 18.</head>
        </head>
      </chunk>
      <chunk pos="191" start="1005476" end="1007561" type="number" n="19" id="elem.5.19" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D19">
        <head lang="en">
          <head>PROPOSITION 19</head>
        </head>
      </chunk>
      <chunk pos="192" start="1007567" end="1009974" type="number" n="20" id="elem.5.20" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D20">
        <head lang="en">
          <head>PROPOSITION 20.</head>
        </head>
      </chunk>
      <chunk pos="193" start="1009980" end="1012453" type="number" n="21" id="elem.5.21" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D21">
        <head lang="en">
          <head>PROPOSITION 21.</head>
        </head>
      </chunk>
      <chunk pos="194" start="1012459" end="1015367" type="number" n="22" id="elem.5.22" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D22">
        <head lang="en">
          <head>PROPOSITION 22.</head>
        </head>
      </chunk>
      <chunk pos="195" start="1015373" end="1020022" type="number" n="23" id="elem.5.23" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D23">
        <head lang="en">
          <head>PROPOSITION 23.</head>
        </head>
      </chunk>
      <chunk pos="196" start="1020028" end="1022282" type="number" n="24" id="elem.5.24" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D24">
        <head lang="en">
          <head>PROPOSITION 24.</head>
        </head>
      </chunk>
      <chunk pos="197" start="1022288" end="1024530" type="number" n="25" id="elem.5.25" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D5%3Atype%3DProp%3Anumber%3D25">
        <head lang="en">
          <head>PROPOSITION 25.</head>
        </head>
      </chunk>
    </chunk>
  </chunk>
  <chunk pos="6" start="1024557" end="1141087" type="book" n="6" id="elem.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6">
    <head lang="en">
      <head>BOOK VI.</head>
    </head>
    <chunk pos="13" start="1024675" end="1025874" type="type" n="Def" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DDef">
      <head lang="en">
        <head>DEFINITIONS.</head>
      </head>
      <chunk pos="198" start="1024790" end="1025047" type="number" n="1" id="elem.6.def.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DDef%3Anumber%3D1">
        <head lang="en">
          <head>1</head>
        </head>
      </chunk>
      <chunk pos="199" start="1025053" end="1025240" type="number" n="2" id="elem.6.def.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DDef%3Anumber%3D2">
        <head lang="en">
          <head>2</head>
        </head>
      </chunk>
      <chunk pos="200" start="1025246" end="1025527" type="number" n="3" id="elem.6.def.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DDef%3Anumber%3D3">
        <head lang="en">
          <head>3</head>
        </head>
      </chunk>
      <chunk pos="201" start="1025533" end="1025746" type="number" n="4" id="elem.6.def.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DDef%3Anumber%3D4">
        <head lang="en">
          <head>4</head>
        </head>
      </chunk>
      <chunk pos="202" start="1025752" end="1025862" type="number" n="5" id="elem.6.def.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DDef%3Anumber%3D5">
        <head lang="en">
          <head>5</head>
        </head>
      </chunk>
    </chunk>
    <chunk pos="14" start="1025879" end="1141076" type="type" n="Prop" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp">
      <head lang="en">
        <head>BOOK VI. PROPOSITIONS.</head>
      </head>
      <chunk pos="203" start="1026005" end="1032028" type="number" n="1" id="elem.6.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D1">
        <head lang="en">
          <head>PROPOSITION 1.</head>
        </head>
      </chunk>
      <chunk pos="204" start="1032034" end="1035556" type="number" n="2" id="elem.6.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D2">
        <head lang="en">
          <head>PROPOSITION 2.</head>
        </head>
      </chunk>
      <chunk pos="205" start="1035562" end="1039642" type="number" n="3" id="elem.6.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D3">
        <head lang="en">
          <head>PROPOSITION 3.</head>
        </head>
      </chunk>
      <chunk pos="206" start="1039648" end="1043315" type="number" n="4" id="elem.6.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D4">
        <head lang="en">
          <head>PROPOSITION 4.</head>
        </head>
      </chunk>
      <chunk pos="207" start="1043321" end="1047528" type="number" n="5" id="elem.6.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D5">
        <head lang="en">
          <head>PROPOSITION 5.</head>
        </head>
      </chunk>
      <chunk pos="208" start="1047534" end="1050996" type="number" n="6" id="elem.6.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D6">
        <head lang="en">
          <head>PROPOSITION 6.</head>
        </head>
      </chunk>
      <chunk pos="209" start="1051002" end="1056115" type="number" n="7" id="elem.6.7" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D7">
        <head lang="en">
          <head>PROPOSITION 7.</head>
        </head>
      </chunk>
      <chunk pos="210" start="1056121" end="1060132" type="number" n="8" id="elem.6.8" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D8">
        <head lang="en">
          <head>PROPOSITION 8.</head>
        </head>
      </chunk>
      <chunk pos="211" start="1060138" end="1062357" type="number" n="9" id="elem.6.9" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D9">
        <head lang="en">
          <head>PROPOSITION 9.</head>
        </head>
      </chunk>
      <chunk pos="212" start="1062363" end="1064658" type="number" n="10" id="elem.6.10" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D10">
        <head lang="en">
          <head>PROPOSITION 10.</head>
        </head>
      </chunk>
      <chunk pos="213" start="1064664" end="1066343" type="number" n="11" id="elem.6.11" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D11">
        <head lang="en">
          <head>PROPOSITION 11.</head>
        </head>
      </chunk>
      <chunk pos="214" start="1066349" end="1067899" type="number" n="12" id="elem.6.12" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D12">
        <head lang="en">
          <head>PROPOSITION 12.</head>
        </head>
      </chunk>
      <chunk pos="215" start="1067905" end="1069204" type="number" n="13" id="elem.6.13" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D13">
        <head lang="en">
          <head>PROPOSITION 13.</head>
        </head>
      </chunk>
      <chunk pos="216" start="1069210" end="1072125" type="number" n="14" id="elem.6.14" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D14">
        <head lang="en">
          <head>PROPOSITION 14.</head>
        </head>
      </chunk>
      <chunk pos="217" start="1072131" end="1075470" type="number" n="15" id="elem.6.15" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D15">
        <head lang="en">
          <head>PROPOSITION 15.</head>
        </head>
      </chunk>
      <chunk pos="218" start="1075476" end="1079130" type="number" n="16" id="elem.6.16" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D16">
        <head lang="en">
          <head>PROPOSITION 16.</head>
        </head>
      </chunk>
      <chunk pos="219" start="1079136" end="1081923" type="number" n="17" id="elem.6.17" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D17">
        <head lang="en">
          <head>PROPOSITION 17</head>
        </head>
      </chunk>
      <chunk pos="220" start="1081929" end="1085586" type="number" n="18" id="elem.6.18" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D18">
        <head lang="en">
          <head>PROPOSITION 18.</head>
        </head>
      </chunk>
      <chunk pos="221" start="1085592" end="1089453" type="number" n="19" id="elem.6.19" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D19">
        <head lang="en">
          <head>PROPOSITION 19.</head>
        </head>
      </chunk>
      <chunk pos="222" start="1089459" end="1099674" type="number" n="20" id="elem.6.20" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D20">
        <head lang="en">
          <head>PROPOSITION 20.</head>
        </head>
      </chunk>
      <chunk pos="223" start="1099680" end="1100761" type="number" n="21" id="elem.6.21" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D21">
        <head lang="en">
          <head>PROPOSITION 21.</head>
        </head>
      </chunk>
      <chunk pos="224" start="1100767" end="1105111" type="number" n="22" id="elem.6.22" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D22">
        <head lang="en">
          <head>PROPOSITION 22.</head>
        </head>
      </chunk>
      <chunk pos="225" start="1105117" end="1109529" type="number" n="23" id="elem.6.23" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D23">
        <head lang="en">
          <head>PROPOSITION 23.</head>
        </head>
      </chunk>
      <chunk pos="226" start="1109535" end="1113815" type="number" n="24" id="elem.6.24" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D24">
        <head lang="en">
          <head>PROPOSITION 24.</head>
        </head>
      </chunk>
      <chunk pos="227" start="1113821" end="1117204" type="number" n="25" id="elem.6.25" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D25">
        <head lang="en">
          <head>PROPOSITION 25.</head>
        </head>
      </chunk>
      <chunk pos="228" start="1117210" end="1119398" type="number" n="26" id="elem.6.26" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D26">
        <head lang="en">
          <head>PROPOSITION 26.</head>
        </head>
      </chunk>
      <chunk pos="229" start="1119404" end="1121761" type="number" n="27" id="elem.6.27" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D27">
        <head lang="en">
          <head>PROPOSITION 27.</head>
        </head>
      </chunk>
      <chunk pos="230" start="1121767" end="1126558" type="number" n="28" id="elem.6.28" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D28">
        <head lang="en">
          <head>PROPOSITION 28.</head>
        </head>
      </chunk>
      <chunk pos="231" start="1126564" end="1130120" type="number" n="29" id="elem.6.29" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D29">
        <head lang="en">
          <head>PROPOSITION 29.</head>
        </head>
      </chunk>
      <chunk pos="232" start="1130126" end="1132075" type="number" n="30" id="elem.6.30" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D30">
        <head lang="en">
          <head>PROPOSITION 30.</head>
        </head>
      </chunk>
      <chunk pos="233" start="1132081" end="1134371" type="number" n="31" id="elem.6.31" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D31">
        <head lang="en">
          <head>PROPOSITION 31.</head>
        </head>
      </chunk>
      <chunk pos="234" start="1134377" end="1137443" type="number" n="32" id="elem.6.32" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D32">
        <head lang="en">
          <head>PROPOSITION 32.</head>
        </head>
      </chunk>
      <chunk pos="235" start="1137449" end="1141064" type="number" n="33" id="elem.6.33" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D6%3Atype%3DProp%3Anumber%3D33">
        <head lang="en">
          <head>PROPOSITION 33.</head>
        </head>
      </chunk>
    </chunk>
  </chunk>
  <chunk pos="7" start="1141109" end="1255043" type="book" n="7" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7">
    <head lang="en">
      <head>BOOK VII.</head>
    </head>
    <chunk pos="15" start="1141197" end="1146612" type="type" n="Def" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef">
      <head lang="en">
        <head>DEFINITIONS.</head>
      </head>
      <chunk pos="236" start="1141292" end="1141504" type="number" n="1" id="elem.7.def.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D1">
        <head lang="en">
          <head>1</head>
        </head>
      </chunk>
      <chunk pos="237" start="1141510" end="1141686" type="number" n="2" id="elem.7.def.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D2">
        <head lang="en">
          <head>2</head>
        </head>
      </chunk>
      <chunk pos="238" start="1141692" end="1141912" type="number" n="3" id="elem.7.def.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D3">
        <head lang="en">
          <head>3</head>
        </head>
      </chunk>
      <chunk pos="239" start="1141918" end="1142090" type="number" n="4" id="elem.7.def.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D4">
        <head lang="en">
          <head>4</head>
        </head>
      </chunk>
      <chunk pos="240" start="1142096" end="1142307" type="number" n="5" id="elem.7.def.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D5">
        <head lang="en">
          <head>5</head>
        </head>
      </chunk>
      <chunk pos="241" start="1142313" end="1142510" type="number" n="6" id="elem.7.def.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D6">
        <head lang="en">
          <head>6</head>
        </head>
      </chunk>
      <chunk pos="242" start="1142516" end="1142770" type="number" n="7" id="elem.7.def.7" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D7">
        <head lang="en">
          <head>7</head>
        </head>
      </chunk>
      <chunk pos="243" start="1142776" end="1143008" type="number" n="8" id="elem.7.def.8" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D8">
        <head lang="en">
          <head>8</head>
        </head>
      </chunk>
      <chunk pos="244" start="1143014" end="1143244" type="number" n="9" id="elem.7.def.9" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D9">
        <head lang="en">
          <head>9</head>
        </head>
      </chunk>
      <chunk pos="245" start="1143250" end="1143481" type="number" n="10" id="elem.7.def.10" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D10">
        <head lang="en">
          <head>10</head>
        </head>
      </chunk>
      <chunk pos="246" start="1143487" end="1143703" type="number" n="11" id="elem.7.def.11" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D11">
        <head lang="en">
          <head>11</head>
        </head>
      </chunk>
      <chunk pos="247" start="1143709" end="1143941" type="number" n="12" id="elem.7.def.12" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D12">
        <head lang="en">
          <head>12</head>
        </head>
      </chunk>
      <chunk pos="248" start="1143947" end="1144144" type="number" n="13" id="elem.7.def.13" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D13">
        <head lang="en">
          <head>13</head>
        </head>
      </chunk>
      <chunk pos="249" start="1144150" end="1144384" type="number" n="14" id="elem.7.def.14" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D14">
        <head lang="en">
          <head>14</head>
        </head>
      </chunk>
      <chunk pos="250" start="1144390" end="1144694" type="number" n="15" id="elem.7.def.15" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D15">
        <head lang="en">
          <head>15</head>
        </head>
      </chunk>
      <chunk pos="251" start="1144700" end="1145032" type="number" n="16" id="elem.7.def.16" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D16">
        <head lang="en">
          <head>16</head>
        </head>
      </chunk>
      <chunk pos="252" start="1145038" end="1145365" type="number" n="17" id="elem.7.def.17" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D17">
        <head lang="en">
          <head>17</head>
        </head>
      </chunk>
      <chunk pos="253" start="1145371" end="1145606" type="number" n="18" id="elem.7.def.18" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D18">
        <head lang="en">
          <head>18</head>
        </head>
      </chunk>
      <chunk pos="254" start="1145612" end="1145863" type="number" n="19" id="elem.7.def.19" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D19">
        <head lang="en">
          <head>19</head>
        </head>
      </chunk>
      <chunk pos="255" start="1145869" end="1146151" type="number" n="20" id="elem.7.def.20" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D20">
        <head lang="en">
          <head>20</head>
        </head>
      </chunk>
      <chunk pos="256" start="1146157" end="1146400" type="number" n="21" id="elem.7.def.21" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D21">
        <head lang="en">
          <head>21</head>
        </head>
      </chunk>
      <chunk pos="257" start="1146406" end="1146600" type="number" n="22" id="elem.7.def.22" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D22">
        <head lang="en">
          <head>22</head>
        </head>
      </chunk>
    </chunk>
    <chunk pos="16" start="1146636" end="1255032" type="type" n="Prop" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp">
      <head lang="en">
        <head>BOOK VII. PROPOSITIONS.</head>
      </head>
      <chunk pos="258" start="1146743" end="1149378" type="number" n="1" id="elem.7.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D1">
        <head lang="en">
          <head>PROPOSITION I.</head>
        </head>
      </chunk>
      <chunk pos="259" start="1149404" end="1153756" type="number" n="2" id="elem.7.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D2">
        <head lang="en">
          <head>PROPOSITION 2.</head>
        </head>
      </chunk>
      <chunk pos="260" start="1153762" end="1159891" type="number" n="3" id="elem.7.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D3">
        <head lang="en">
          <head>PROPOSITION 3.</head>
        </head>
      </chunk>
      <chunk pos="261" start="1159897" end="1162059" type="number" n="4" id="elem.7.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D4">
        <head lang="en">
          <head>PROPOSITION 4.</head>
        </head>
      </chunk>
      <chunk pos="262" start="1162085" end="1164829" type="number" n="5" id="elem.7.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D5">
        <head lang="en">
          <head>PROPOSITION 5.</head>
        </head>
      </chunk>
      <chunk pos="263" start="1164855" end="1167369" type="number" n="6" id="elem.7.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D6">
        <head lang="en">
          <head>PROPOSITION 6.</head>
        </head>
      </chunk>
      <chunk pos="264" start="1167375" end="1169976" type="number" n="7" id="elem.7.7" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D7">
        <head lang="en">
          <head>PROPOSITION 7.</head>
        </head>
      </chunk>
      <chunk pos="265" start="1169982" end="1173868" type="number" n="8" id="elem.7.8" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D8">
        <head lang="en">
          <head>PROPOSITION 8.</head>
        </head>
      </chunk>
      <chunk pos="266" start="1173874" end="1176658" type="number" n="9" id="elem.7.9" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D9">
        <head lang="en">
          <head>PROPOSITION 9.</head>
        </head>
      </chunk>
      <chunk pos="267" start="1176664" end="1179125" type="number" n="10" id="elem.7.10" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D10">
        <head lang="en">
          <head>PROPOSITION 10.</head>
        </head>
      </chunk>
      <chunk pos="268" start="1179131" end="1180633" type="number" n="11" id="elem.7.11" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D11">
        <head lang="en">
          <head>PROPOSITION 11.</head>
        </head>
      </chunk>
      <chunk pos="269" start="1180639" end="1182383" type="number" n="12" id="elem.7.12" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D12">
        <head lang="en">
          <head>PROPOSITION 12.</head>
        </head>
      </chunk>
      <chunk pos="270" start="1182409" end="1184018" type="number" n="13" id="elem.7.13" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D13">
        <head lang="en">
          <head>PROPOSITION 13.</head>
        </head>
      </chunk>
      <chunk pos="271" start="1184024" end="1186301" type="number" n="14" id="elem.7.14" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D14">
        <head lang="en">
          <head>PROPOSITION 14.</head>
        </head>
      </chunk>
      <chunk pos="272" start="1186307" end="1189611" type="number" n="15" id="elem.7.15" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D15">
        <head lang="en">
          <head>PROPOSITION 15.</head>
        </head>
      </chunk>
      <chunk pos="273" start="1189637" end="1192508" type="number" n="16" id="elem.7.16" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D16">
        <head lang="en">
          <head>PROPOSITION 16.</head>
        </head>
      </chunk>
      <chunk pos="274" start="1192514" end="1194415" type="number" n="17" id="elem.7.17" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D17">
        <head lang="en">
          <head>PROPOSITION 17.</head>
        </head>
      </chunk>
      <chunk pos="275" start="1194441" end="1195852" type="number" n="18" id="elem.7.18" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D18">
        <head lang="en">
          <head>PROPOSITION 18.</head>
        </head>
      </chunk>
      <chunk pos="276" start="1195858" end="1200024" type="number" n="19" id="elem.7.19" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D19">
        <head lang="en">
          <head>PROPOSITION 19.</head>
        </head>
      </chunk>
      <chunk pos="277" start="1200030" end="1203488" type="number" n="20" id="elem.7.20" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D20">
        <head lang="en">
          <head>PROPOSITION 20.</head>
        </head>
      </chunk>
      <chunk pos="278" start="1203494" end="1206066" type="number" n="21" id="elem.7.21" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D21">
        <head lang="en">
          <head>PROPOSITION 21.</head>
        </head>
      </chunk>
      <chunk pos="279" start="1206072" end="1208209" type="number" n="22" id="elem.7.22" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D22">
        <head lang="en">
          <head>PROPOSITION 22.</head>
        </head>
      </chunk>
      <chunk pos="280" start="1208215" end="1209736" type="number" n="23" id="elem.7.23" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D23">
        <head lang="en">
          <head>PROPOSITION 23.</head>
        </head>
      </chunk>
      <chunk pos="281" start="1209742" end="1213329" type="number" n="24" id="elem.7.24" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D24">
        <head lang="en">
          <head>PROPOSITION 24.</head>
        </head>
      </chunk>
      <chunk pos="282" start="1213335" end="1215106" type="number" n="25" id="elem.7.25" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D25">
        <head lang="en">
          <head>PROPOSITION 25.</head>
        </head>
      </chunk>
      <chunk pos="283" start="1215112" end="1216986" type="number" n="26" id="elem.7.26" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D26">
        <head lang="en">
          <head>PROPOSITION 26.</head>
        </head>
      </chunk>
      <chunk pos="284" start="1217012" end="1219650" type="number" n="27" id="elem.7.27" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D27">
        <head lang="en">
          <head>PROPOSITION 27.</head>
        </head>
      </chunk>
      <chunk pos="285" start="1219656" end="1222550" type="number" n="28" id="elem.7.28" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D28">
        <head lang="en">
          <head>PROPOSITION 28.</head>
        </head>
      </chunk>
      <chunk pos="286" start="1222556" end="1223850" type="number" n="29" id="elem.7.29" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D29">
        <head lang="en">
          <head>PROPOSITION 29.</head>
        </head>
      </chunk>
      <chunk pos="287" start="1223856" end="1226471" type="number" n="30" id="elem.7.30" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D30">
        <head lang="en">
          <head>PROPOSITION 30.</head>
        </head>
      </chunk>
      <chunk pos="288" start="1226477" end="1229020" type="number" n="31" id="elem.7.31" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D31">
        <head lang="en">
          <head>PROPOSITION 31.</head>
        </head>
      </chunk>
      <chunk pos="289" start="1229026" end="1229741" type="number" n="32" id="elem.7.32" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D32">
        <head lang="en">
          <head>PROPOSITION 32.</head>
        </head>
      </chunk>
      <chunk pos="290" start="1229747" end="1235997" type="number" n="33" id="elem.7.33" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D33">
        <head lang="en">
          <head>PROPOSITION 33.</head>
        </head>
      </chunk>
      <chunk pos="291" start="1236023" end="1242766" type="number" n="34" id="elem.7.34" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D34">
        <head lang="en">
          <head>PROPOSITION 34.</head>
        </head>
      </chunk>
      <chunk pos="292" start="1242792" end="1244140" type="number" n="35" id="elem.7.35" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D35">
        <head lang="en">
          <head>PROPOSITION 35.</head>
        </head>
      </chunk>
      <chunk pos="293" start="1244146" end="1248883" type="number" n="36" id="elem.7.36" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D36">
        <head lang="en">
          <head>PROPOSITION 36.</head>
        </head>
      </chunk>
      <chunk pos="294" start="1248889" end="1250788" type="number" n="37" id="elem.7.37" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D37">
        <head lang="en">
          <head>PROPOSITION 37.</head>
        </head>
      </chunk>
      <chunk pos="295" start="1250794" end="1252300" type="number" n="38" id="elem.7.38" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D38">
        <head lang="en">
          <head>PROPOSITION 38.</head>
        </head>
      </chunk>
      <chunk pos="296" start="1252306" end="1255020" type="number" n="39" id="elem.7.39" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DProp%3Anumber%3D39">
        <head lang="en">
          <head>PROPOSITION 39.</head>
        </head>
      </chunk>
    </chunk>
  </chunk>
  <chunk pos="8" start="1255047" end="1358298" type="book" n="8" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8">
    <head lang="en">
      <head>BOOK VIII.</head>
    </head>
    <chunk pos="17" start="1255155" end="1358287" type="type" n="Prop" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp">
      <head lang="en">
        <head>PROPOSITIONS</head>
      </head>
      <chunk pos="297" start="1255251" end="1258095" type="number" n="1" id="elem.8.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D1">
        <head lang="en">
          <head>PROPOSITION 1.</head>
        </head>
      </chunk>
      <chunk pos="298" start="1258101" end="1264550" type="number" n="2" id="elem.8.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D2">
        <head lang="en">
          <head>PROPOSITION 2.</head>
        </head>
      </chunk>
      <chunk pos="299" start="1264556" end="1268119" type="number" n="3" id="elem.8.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D3">
        <head lang="en">
          <head>PROPOSITION 3.</head>
        </head>
      </chunk>
      <chunk pos="300" start="1268145" end="1278773" type="number" n="4" id="elem.8.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D4">
        <head lang="en">
          <head>PROPOSITION 4.</head>
        </head>
      </chunk>
      <chunk pos="301" start="1278799" end="1282606" type="number" n="5" id="elem.8.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D5">
        <head lang="en">
          <head>PROPOSITION 5.</head>
        </head>
      </chunk>
      <chunk pos="302" start="1282612" end="1285606" type="number" n="6" id="elem.8.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D6">
        <head lang="en">
          <head>PROPOSITION 6.</head>
        </head>
      </chunk>
      <chunk pos="303" start="1285612" end="1286604" type="number" n="7" id="elem.8.7" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D7">
        <head lang="en">
          <head>PROPOSITION 7.</head>
        </head>
      </chunk>
      <chunk pos="304" start="1286630" end="1291899" type="number" n="8" id="elem.8.8" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D8">
        <head lang="en">
          <head>PROPOSITION 8.</head>
        </head>
      </chunk>
      <chunk pos="305" start="1291905" end="1297695" type="number" n="9" id="elem.8.9" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D9">
        <head lang="en">
          <head>PROPOSITION 9.</head>
        </head>
      </chunk>
      <chunk pos="306" start="1297701" end="1304135" type="number" n="10" id="elem.8.10" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D10">
        <head lang="en">
          <head>PROPOSITION 10.</head>
        </head>
      </chunk>
      <chunk pos="307" start="1304141" end="1307055" type="number" n="11" id="elem.8.11" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D11">
        <head lang="en">
          <head>PROPOSITION 11.</head>
        </head>
      </chunk>
      <chunk pos="308" start="1307061" end="1312186" type="number" n="12" id="elem.8.12" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D12">
        <head lang="en">
          <head>PROPOSITION 12.</head>
        </head>
      </chunk>
      <chunk pos="309" start="1312192" end="1316009" type="number" n="13" id="elem.8.13" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D13">
        <head lang="en">
          <head>PROPOSITION 13.</head>
        </head>
      </chunk>
      <chunk pos="310" start="1316015" end="1318529" type="number" n="14" id="elem.8.14" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D14">
        <head lang="en">
          <head>PROPOSITION 14.</head>
        </head>
      </chunk>
      <chunk pos="311" start="1318535" end="1321411" type="number" n="15" id="elem.8.15" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D15">
        <head lang="en">
          <head>PROPOSITION 15.</head>
        </head>
      </chunk>
      <chunk pos="312" start="1321437" end="1322990" type="number" n="16" id="elem.8.16" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D16">
        <head lang="en">
          <head>PROPOSITION 16.</head>
        </head>
      </chunk>
      <chunk pos="313" start="1322996" end="1324565" type="number" n="17" id="elem.8.17" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D17">
        <head lang="en">
          <head>PROPOSITION 17.</head>
        </head>
      </chunk>
      <chunk pos="314" start="1324571" end="1329321" type="number" n="18" id="elem.8.18" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D18">
        <head lang="en">
          <head>PROPOSITION 18.</head>
        </head>
      </chunk>
      <chunk pos="315" start="1329347" end="1338634" type="number" n="19" id="elem.8.19" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D19">
        <head lang="en">
          <head>PROPOSITION 19.</head>
        </head>
      </chunk>
      <chunk pos="316" start="1338640" end="1343817" type="number" n="20" id="elem.8.20" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D20">
        <head lang="en">
          <head>PROPOSITION 20.</head>
        </head>
      </chunk>
      <chunk pos="317" start="1343823" end="1349943" type="number" n="21" id="elem.8.21" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D21">
        <head lang="en">
          <head>PROPOSITION 21.</head>
        </head>
      </chunk>
      <chunk pos="318" start="1349949" end="1350872" type="number" n="22" id="elem.8.22" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D22">
        <head lang="en">
          <head>PROPOSITION 22.</head>
        </head>
      </chunk>
      <chunk pos="319" start="1350878" end="1351831" type="number" n="23" id="elem.8.23" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D23">
        <head lang="en">
          <head>PROPOSITION 23.</head>
        </head>
      </chunk>
      <chunk pos="320" start="1351837" end="1353258" type="number" n="24" id="elem.8.24" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D24">
        <head lang="en">
          <head>PROPOSITION 24.</head>
        </head>
      </chunk>
      <chunk pos="321" start="1353264" end="1354969" type="number" n="25" id="elem.8.25" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D25">
        <head lang="en">
          <head>PROPOSITION 25.</head>
        </head>
      </chunk>
      <chunk pos="322" start="1354975" end="1356617" type="number" n="26" id="elem.8.26" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D26">
        <head lang="en">
          <head>PROPOSITION 26.</head>
        </head>
      </chunk>
      <chunk pos="323" start="1356623" end="1358275" type="number" n="27" id="elem.8.27" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D8%3Atype%3DProp%3Anumber%3D27">
        <head lang="en">
          <head>PROPOSITION 27.</head>
        </head>
      </chunk>
    </chunk>
  </chunk>
  <chunk pos="9" start="1358302" end="1453500" type="book" n="9" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9">
    <head lang="en">
      <head>BOOK IX.</head>
    </head>
    <chunk pos="18" start="1358408" end="1453489" type="type" n="Prop" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp">
      <head lang="en">
        <head>PROPOSITIONS</head>
      </head>
      <chunk pos="324" start="1358504" end="1360283" type="number" n="1" id="elem.9.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D1">
        <head lang="en">
          <head>PROPOSITION 1.</head>
        </head>
      </chunk>
      <chunk pos="325" start="1360309" end="1362266" type="number" n="2" id="elem.9.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D2">
        <head lang="en">
          <head>PROPOSITION 2.</head>
        </head>
      </chunk>
      <chunk pos="326" start="1362272" end="1365439" type="number" n="3" id="elem.9.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D3">
        <head lang="en">
          <head>PROPOSITION 3.</head>
        </head>
      </chunk>
      <chunk pos="327" start="1365465" end="1367125" type="number" n="4" id="elem.9.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D4">
        <head lang="en">
          <head>PROPOSITION 4.</head>
        </head>
      </chunk>
      <chunk pos="328" start="1367131" end="1368900" type="number" n="5" id="elem.9.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D5">
        <head lang="en">
          <head>PROPOSITION 5.</head>
        </head>
      </chunk>
      <chunk pos="329" start="1368906" end="1371696" type="number" n="6" id="elem.9.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D6">
        <head lang="en">
          <head>PROPOSITION 6.</head>
        </head>
      </chunk>
      <chunk pos="330" start="1371702" end="1373342" type="number" n="7" id="elem.9.7" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D7">
        <head lang="en">
          <head>PROPOSITION 7.</head>
        </head>
      </chunk>
      <chunk pos="331" start="1373348" end="1377066" type="number" n="8" id="elem.9.8" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D8">
        <head lang="en">
          <head>PROPOSITION 8.</head>
        </head>
      </chunk>
      <chunk pos="332" start="1377072" end="1380245" type="number" n="9" id="elem.9.9" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D9">
        <head lang="en">
          <head>PROPOSITION 9.</head>
        </head>
      </chunk>
      <chunk pos="333" start="1380251" end="1383993" type="number" n="10" id="elem.9.10" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D10">
        <head lang="en">
          <head>PROPOSITION 10.</head>
        </head>
      </chunk>
      <chunk pos="334" start="1383999" end="1386258" type="number" n="11" id="elem.9.11" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D11">
        <head lang="en">
          <head>PROPOSITION II.</head>
        </head>
      </chunk>
      <chunk pos="335" start="1386284" end="1392175" type="number" n="12" id="elem.9.12" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D12">
        <head lang="en">
          <head>PROPOSITION 12.</head>
        </head>
      </chunk>
      <chunk pos="336" start="1392181" end="1400412" type="number" n="13" id="elem.9.13" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D13">
        <head lang="en">
          <head>PROPOSITION 13.</head>
        </head>
      </chunk>
      <chunk pos="337" start="1400418" end="1402805" type="number" n="14" id="elem.9.14" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D14">
        <head lang="en">
          <head>PROPOSITION 14.</head>
        </head>
      </chunk>
      <chunk pos="338" start="1402831" end="1408222" type="number" n="15" id="elem.9.15" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D15">
        <head lang="en">
          <head>PROPOSITION 15.</head>
        </head>
      </chunk>
      <chunk pos="339" start="1408248" end="1409707" type="number" n="16" id="elem.9.16" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D16">
        <head lang="en">
          <head>PROPOSITION 16.</head>
        </head>
      </chunk>
      <chunk pos="340" start="1409713" end="1412305" type="number" n="17" id="elem.9.17" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D17">
        <head lang="en">
          <head>PROPOSITION 17.</head>
        </head>
      </chunk>
      <chunk pos="341" start="1412311" end="1415234" type="number" n="18" id="elem.9.18" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D18">
        <head lang="en">
          <head>PROPOSITION 18.</head>
        </head>
      </chunk>
      <chunk pos="342" start="1415260" end="1421480" type="number" n="19" id="elem.9.19" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D19">
        <head lang="en">
          <head>PROPOSITION 19.</head>
        </head>
      </chunk>
      <chunk pos="343" start="1421486" end="1423905" type="number" n="20" id="elem.9.20" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D20">
        <head lang="en">
          <head>PROPOSITION 20.</head>
        </head>
      </chunk>
      <chunk pos="344" start="1423911" end="1424862" type="number" n="21" id="elem.9.21" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D21">
        <head lang="en">
          <head>PROPOSITION 21.</head>
        </head>
      </chunk>
      <chunk pos="345" start="1424868" end="1425995" type="number" n="22" id="elem.9.22" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D22">
        <head lang="en">
          <head>PROPOSITION 22.</head>
        </head>
      </chunk>
      <chunk pos="346" start="1426001" end="1427418" type="number" n="23" id="elem.9.23" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D23">
        <head lang="en">
          <head>PROPOSITION 23.</head>
        </head>
      </chunk>
      <chunk pos="347" start="1427424" end="1428231" type="number" n="24" id="elem.9.24" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D24">
        <head lang="en">
          <head>PROPOSITION 24.</head>
        </head>
      </chunk>
      <chunk pos="348" start="1428237" end="1429221" type="number" n="25" id="elem.9.25" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D25">
        <head lang="en">
          <head>PROPOSITION 25.</head>
        </head>
      </chunk>
      <chunk pos="349" start="1429227" end="1430153" type="number" n="26" id="elem.9.26" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D26">
        <head lang="en">
          <head>PROPOSITION 26.</head>
        </head>
      </chunk>
      <chunk pos="350" start="1430179" end="1431146" type="number" n="27" id="elem.9.27" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D27">
        <head lang="en">
          <head>PROPOSITION 27.</head>
        </head>
      </chunk>
      <chunk pos="351" start="1431152" end="1432262" type="number" n="28" id="elem.9.28" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D28">
        <head lang="en">
          <head>PROPOSITION 28.</head>
        </head>
      </chunk>
      <chunk pos="352" start="1432268" end="1433360" type="number" n="29" id="elem.9.29" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D29">
        <head lang="en">
          <head>PROPOSITION 29.</head>
        </head>
      </chunk>
      <chunk pos="353" start="1433366" end="1434773" type="number" n="30" id="elem.9.30" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D30">
        <head lang="en">
          <head>PROPOSITION 30.</head>
        </head>
      </chunk>
      <chunk pos="354" start="1434779" end="1436385" type="number" n="31" id="elem.9.31" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D31">
        <head lang="en">
          <head>PROPOSITION 31.</head>
        </head>
      </chunk>
      <chunk pos="355" start="1436391" end="1438155" type="number" n="32" id="elem.9.32" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D32">
        <head lang="en">
          <head>PROPOSITION 32.</head>
        </head>
      </chunk>
      <chunk pos="356" start="1438161" end="1439163" type="number" n="33" id="elem.9.33" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D33">
        <head lang="en">
          <head>PROPOSITION 33.</head>
        </head>
      </chunk>
      <chunk pos="357" start="1439169" end="1440524" type="number" n="34" id="elem.9.34" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D34">
        <head lang="en">
          <head>PROPOSITION 34.</head>
        </head>
      </chunk>
      <chunk pos="358" start="1440530" end="1443848" type="number" n="35" id="elem.9.35" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D35">
        <head lang="en">
          <head>PROPOSITION 35.</head>
        </head>
      </chunk>
      <chunk pos="359" start="1443854" end="1453477" type="number" n="36" id="elem.9.36" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D9%3Atype%3DProp%3Anumber%3D36">
        <head lang="en">
          <head>PROPOSITION 36.</head>
        </head>
      </chunk>
    </chunk>
  </chunk>
  <chunk pos="10" start="1453541" end="1953976" type="book" n="10" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10">
    <head lang="en">
      <head>BOOK X.</head>
    </head>
    <chunk pos="19" start="1453645" end="1455683" type="type" n="Def 1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DDef+1">
      <head lang="en">
        <head>DEFINITIONS I.</head>
      </head>
      <chunk pos="360" start="1453744" end="1454048" type="number" n="1" id="elem.10.def.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DDef+1%3Anumber%3D1">
        <head lang="en">
          <head>1</head>
        </head>
      </chunk>
      <chunk pos="361" start="1454054" end="1454413" type="number" n="2" id="elem.10.def.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DDef+1%3Anumber%3D2">
        <head lang="en">
          <head>2</head>
        </head>
      </chunk>
      <chunk pos="362" start="1454419" end="1455054" type="number" n="3" id="elem.10.def.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DDef+1%3Anumber%3D3">
        <head lang="en">
          <head>3</head>
        </head>
      </chunk>
      <chunk pos="363" start="1455060" end="1455671" type="number" n="4" id="elem.10.def.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DDef+1%3Anumber%3D4">
        <head lang="en">
          <head>4</head>
        </head>
      </chunk>
    </chunk>
    <chunk pos="20" start="1455724" end="1631234" type="type" n="Prop 1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1">
      <head lang="en">
        <head>PROPOSITIONS I?47.</head>
      </head>
      <chunk pos="364" start="1455830" end="1459087" type="number" n="1" id="elem.10.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D1">
        <head lang="en">
          <head>PROPOSITION 1.</head>
        </head>
      </chunk>
      <chunk pos="365" start="1459112" end="1461498" type="number" n="2" id="elem.10.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D2">
        <head lang="en">
          <head>PROPOSITION 2.</head>
        </head>
      </chunk>
      <chunk pos="366" start="1461504" end="1465550" type="number" n="3" id="elem.10.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D3">
        <head lang="en">
          <head>PROPOSITION 3.</head>
        </head>
      </chunk>
      <chunk pos="367" start="1465556" end="1470777" type="number" n="4" id="elem.10.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D4">
        <head lang="en">
          <head>PROPOSITION 4.</head>
        </head>
      </chunk>
      <chunk pos="368" start="1470783" end="1473557" type="number" n="5" id="elem.10.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D5">
        <head lang="en">
          <head>PROPOSITION 5.</head>
        </head>
      </chunk>
      <chunk pos="369" start="1473582" end="1478872" type="number" n="6" id="elem.10.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D6">
        <head lang="en">
          <head>PROPOSITION 6.</head>
        </head>
      </chunk>
      <chunk pos="370" start="1478897" end="1479776" type="number" n="7" id="elem.10.7" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D7">
        <head lang="en">
          <head>PROPOSITION 7.</head>
        </head>
      </chunk>
      <chunk pos="371" start="1479782" end="1480696" type="number" n="8" id="elem.10.8" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D8">
        <head lang="en">
          <head>PROPOSITION 8.</head>
        </head>
      </chunk>
      <chunk pos="372" start="1480702" end="1487089" type="number" n="9" id="elem.10.9" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D9">
        <head lang="en">
          <head>PROPOSITION 9.</head>
        </head>
      </chunk>
      <chunk pos="373" start="1487095" end="1489718" type="number" n="10" id="elem.10.10" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D10">
        <head lang="en">
          <head>[PROPOSITION 10.</head>
        </head>
      </chunk>
      <chunk pos="374" start="1489724" end="1492057" type="number" n="11" id="elem.10.11" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D11">
        <head lang="en">
          <head>PROPOSITION 11.</head>
        </head>
      </chunk>
      <chunk pos="375" start="1492063" end="1495429" type="number" n="12" id="elem.10.12" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D12">
        <head lang="en">
          <head>PROPOSITION 12.</head>
        </head>
      </chunk>
      <chunk pos="376" start="1495454" end="1498708" type="number" n="13" id="elem.10.13" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D13">
        <head lang="en">
          <head>PROPOSITION 13.</head>
        </head>
      </chunk>
      <chunk pos="377" start="1498714" end="1503288" type="number" n="14" id="elem.10.14" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D14">
        <head lang="en">
          <head>PROPOSITION 14.</head>
        </head>
      </chunk>
      <chunk pos="378" start="1503313" end="1505590" type="number" n="15" id="elem.10.15" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D15">
        <head lang="en">
          <head>PROPOSITION 15.</head>
        </head>
      </chunk>
      <chunk pos="379" start="1505615" end="1509793" type="number" n="16" id="elem.10.16" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D16">
        <head lang="en">
          <head>PROPOSITION 16.</head>
        </head>
      </chunk>
      <chunk pos="380" start="1509799" end="1516919" type="number" n="17" id="elem.10.17" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D17">
        <head lang="en">
          <head>PROPOSITION 17.</head>
        </head>
      </chunk>
      <chunk pos="381" start="1516925" end="1522991" type="number" n="18" id="elem.10.18" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D18">
        <head lang="en">
          <head>PROPOSITION 18.</head>
        </head>
      </chunk>
      <chunk pos="382" start="1522997" end="1524497" type="number" n="19" id="elem.10.19" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D19">
        <head lang="en">
          <head>PROPOSITION 19.</head>
        </head>
      </chunk>
      <chunk pos="383" start="1524503" end="1526213" type="number" n="20" id="elem.10.20" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D20">
        <head lang="en">
          <head>PROPOSITION 20.</head>
        </head>
      </chunk>
      <chunk pos="384" start="1526219" end="1529801" type="number" n="21" id="elem.10.21" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D21">
        <head lang="en">
          <head>PROPOSITION 21.</head>
        </head>
      </chunk>
      <chunk pos="385" start="1529807" end="1533751" type="number" n="22" id="elem.10.22" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D22">
        <head lang="en">
          <head>PROPOSITION 22.</head>
        </head>
      </chunk>
      <chunk pos="386" start="1533757" end="1537574" type="number" n="23" id="elem.10.23" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D23">
        <head lang="en">
          <head>PROPOSITION 23.</head>
        </head>
      </chunk>
      <chunk pos="387" start="1537580" end="1538844" type="number" n="24" id="elem.10.24" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D24">
        <head lang="en">
          <head>PROPOSITION 24.</head>
        </head>
      </chunk>
      <chunk pos="388" start="1538850" end="1544143" type="number" n="25" id="elem.10.25" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D25">
        <head lang="en">
          <head>PROPOSITION 25.</head>
        </head>
      </chunk>
      <chunk pos="389" start="1544149" end="1548186" type="number" n="26" id="elem.10.26" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D26">
        <head lang="en">
          <head>PROPOSITION 26.</head>
        </head>
      </chunk>
      <chunk pos="390" start="1548192" end="1551168" type="number" n="27" id="elem.10.27" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D27">
        <head lang="en">
          <head>PROPOSITION 27.</head>
        </head>
      </chunk>
      <chunk pos="391" start="1551174" end="1561410" type="number" n="28" id="elem.10.28" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D28">
        <head lang="en">
          <head>PROPOSITION 28.</head>
        </head>
      </chunk>
      <chunk pos="392" start="1561416" end="1565530" type="number" n="29" id="elem.10.29" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D29">
        <head lang="en">
          <head>PROPOSITION 29.</head>
        </head>
      </chunk>
      <chunk pos="393" start="1565536" end="1568268" type="number" n="30" id="elem.10.30" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D30">
        <head lang="en">
          <head>PROPOSITION 30.</head>
        </head>
      </chunk>
      <chunk pos="394" start="1568274" end="1572528" type="number" n="31" id="elem.10.31" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D31">
        <head lang="en">
          <head>PROPOSITION 31.</head>
        </head>
      </chunk>
      <chunk pos="395" start="1572534" end="1580282" type="number" n="32" id="elem.10.32" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D32">
        <head lang="en">
          <head>PROPOSITION 32.</head>
        </head>
      </chunk>
      <chunk pos="396" start="1580288" end="1584770" type="number" n="33" id="elem.10.33" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D33">
        <head lang="en">
          <head>PROPOSITION 33.</head>
        </head>
      </chunk>
      <chunk pos="397" start="1584776" end="1588262" type="number" n="34" id="elem.10.34" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D34">
        <head lang="en">
          <head>PROPOSITION 34.</head>
        </head>
      </chunk>
      <chunk pos="398" start="1588268" end="1592236" type="number" n="35" id="elem.10.35" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D35">
        <head lang="en">
          <head>PROPOSITION 35.</head>
        </head>
      </chunk>
      <chunk pos="399" start="1592242" end="1594931" type="number" n="36" id="elem.10.36" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D36">
        <head lang="en">
          <head>PROPOSITION 36.</head>
        </head>
      </chunk>
      <chunk pos="400" start="1594937" end="1596875" type="number" n="37" id="elem.10.37" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D37">
        <head lang="en">
          <head>PROPOSITION 37.</head>
        </head>
      </chunk>
      <chunk pos="401" start="1596881" end="1602016" type="number" n="38" id="elem.10.38" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D38">
        <head lang="en">
          <head>PROPOSITION 38.</head>
        </head>
      </chunk>
      <chunk pos="402" start="1602022" end="1603904" type="number" n="39" id="elem.10.39" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D39">
        <head lang="en">
          <head>PROPOSITION 39.</head>
        </head>
      </chunk>
      <chunk pos="403" start="1603910" end="1605682" type="number" n="40" id="elem.10.40" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D40">
        <head lang="en">
          <head>PROPOSITION 40.</head>
        </head>
      </chunk>
      <chunk pos="404" start="1605688" end="1612221" type="number" n="41" id="elem.10.41" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D41">
        <head lang="en">
          <head>PROPOSITION 41.</head>
        </head>
      </chunk>
      <chunk pos="405" start="1612227" end="1615410" type="number" n="42" id="elem.10.42" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D42">
        <head lang="en">
          <head>PROPOSITION 42.</head>
        </head>
      </chunk>
      <chunk pos="406" start="1615416" end="1617200" type="number" n="43" id="elem.10.43" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D43">
        <head lang="en">
          <head>PROPOSITION 43.</head>
        </head>
      </chunk>
      <chunk pos="407" start="1617206" end="1623533" type="number" n="44" id="elem.10.44" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D44">
        <head lang="en">
          <head>PROPOSITION 44.</head>
        </head>
      </chunk>
      <chunk pos="408" start="1623558" end="1625554" type="number" n="45" id="elem.10.45" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D45">
        <head lang="en">
          <head>PROPOSITION 45.</head>
        </head>
      </chunk>
      <chunk pos="409" start="1625579" end="1627631" type="number" n="46" id="elem.10.46" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D46">
        <head lang="en">
          <head>PROPOSITION 46.</head>
        </head>
      </chunk>
      <chunk pos="410" start="1627637" end="1631222" type="number" n="47" id="elem.10.47" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+1%3Anumber%3D47">
        <head lang="en">
          <head>PROPOSITION 47.</head>
        </head>
      </chunk>
    </chunk>
    <chunk pos="21" start="1631239" end="1633224" type="type" n="Def 2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DDef+2">
      <head lang="en">
        <head>DEFINITIONS II.</head>
      </head>
      <chunk pos="411" start="1631339" end="1631852" type="number" n="1" id="elem.10.def.2.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DDef+2%3Anumber%3D1">
        <head lang="en">
          <head>1</head>
        </head>
      </chunk>
      <chunk pos="412" start="1631858" end="1632127" type="number" n="2" id="elem.10.def.2.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DDef+2%3Anumber%3D2">
        <head lang="en">
          <head>2</head>
        </head>
      </chunk>
      <chunk pos="413" start="1632153" end="1632426" type="number" n="3" id="elem.10.def.2.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DDef+2%3Anumber%3D3">
        <head lang="en">
          <head>3</head>
        </head>
      </chunk>
      <chunk pos="414" start="1632432" end="1632863" type="number" n="4" id="elem.10.def.2.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DDef+2%3Anumber%3D4">
        <head lang="en">
          <head>4</head>
        </head>
      </chunk>
      <chunk pos="415" start="1632869" end="1633037" type="number" n="5" id="elem.10.def.2.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DDef+2%3Anumber%3D5">
        <head lang="en">
          <head>5</head>
        </head>
      </chunk>
      <chunk pos="416" start="1633043" end="1633212" type="number" n="6" id="elem.10.def.2.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DDef+2%3Anumber%3D6">
        <head lang="en">
          <head>6</head>
        </head>
      </chunk>
    </chunk>
    <chunk pos="22" start="1633229" end="1790118" type="type" n="Prop 2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2">
      <head lang="en">
        <head>PROPOSITIONS 48?84.</head>
      </head>
      <chunk pos="417" start="1633336" end="1637479" type="number" n="48" id="elem.10.48" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D48">
        <head lang="en">
          <head>PROPOSITION 48.</head>
        </head>
      </chunk>
      <chunk pos="418" start="1637485" end="1641213" type="number" n="49" id="elem.10.49" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D49">
        <head lang="en">
          <head>PROPOSITION 49.</head>
        </head>
      </chunk>
      <chunk pos="419" start="1641219" end="1646861" type="number" n="50" id="elem.10.50" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D50">
        <head lang="en">
          <head>PROPOSITION 50.</head>
        </head>
      </chunk>
      <chunk pos="420" start="1646867" end="1650328" type="number" n="51" id="elem.10.51" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D51">
        <head lang="en">
          <head>PROPOSITION 51.</head>
        </head>
      </chunk>
      <chunk pos="421" start="1650334" end="1653661" type="number" n="52" id="elem.10.52" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D52">
        <head lang="en">
          <head>PROPOSITION 52.</head>
        </head>
      </chunk>
      <chunk pos="422" start="1653667" end="1663144" type="number" n="53" id="elem.10.53" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D53">
        <head lang="en">
          <head>PROPOSITION 53.</head>
        </head>
      </chunk>
      <chunk pos="423" start="1663150" end="1671204" type="number" n="54" id="elem.10.54" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D54">
        <head lang="en">
          <head>PROPOSITION 54.</head>
        </head>
      </chunk>
      <chunk pos="424" start="1671210" end="1678989" type="number" n="55" id="elem.10.55" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D55">
        <head lang="en">
          <head>PROPOSITION 55.</head>
        </head>
      </chunk>
      <chunk pos="425" start="1679015" end="1681812" type="number" n="56" id="elem.10.56" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D56">
        <head lang="en">
          <head>PROPOSITION 56.</head>
        </head>
      </chunk>
      <chunk pos="426" start="1681818" end="1686188" type="number" n="57" id="elem.10.57" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D57">
        <head lang="en">
          <head>PROPOSITION 57.</head>
        </head>
      </chunk>
      <chunk pos="427" start="1686214" end="1689658" type="number" n="58" id="elem.10.58" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D58">
        <head lang="en">
          <head>PROPOSITION 58.</head>
        </head>
      </chunk>
      <chunk pos="428" start="1689684" end="1694372" type="number" n="59" id="elem.10.59" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D59">
        <head lang="en">
          <head>PROPOSITION 59.</head>
        </head>
      </chunk>
      <chunk pos="429" start="1694378" end="1700974" type="number" n="60" id="elem.10.60" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D60">
        <head lang="en">
          <head>PROPOSITION 60.</head>
        </head>
      </chunk>
      <chunk pos="430" start="1700980" end="1705020" type="number" n="61" id="elem.10.61" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D61">
        <head lang="en">
          <head>PROPOSITION 61.</head>
        </head>
      </chunk>
      <chunk pos="431" start="1705026" end="1709160" type="number" n="62" id="elem.10.62" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D62">
        <head lang="en">
          <head>PROPOSITION 62.</head>
        </head>
      </chunk>
      <chunk pos="432" start="1709166" end="1713247" type="number" n="63" id="elem.10.63" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D63">
        <head lang="en">
          <head>PROPOSITION 63.</head>
        </head>
      </chunk>
      <chunk pos="433" start="1713273" end="1716365" type="number" n="64" id="elem.10.64" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D64">
        <head lang="en">
          <head>PROPOSITION 64.</head>
        </head>
      </chunk>
      <chunk pos="434" start="1716371" end="1719624" type="number" n="65" id="elem.10.65" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D65">
        <head lang="en">
          <head>PROPOSITION 65.</head>
        </head>
      </chunk>
      <chunk pos="435" start="1719630" end="1725885" type="number" n="66" id="elem.10.66" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D66">
        <head lang="en">
          <head>PROPOSITION 66.</head>
        </head>
      </chunk>
      <chunk pos="436" start="1725891" end="1729854" type="number" n="67" id="elem.10.67" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D67">
        <head lang="en">
          <head>PROPOSITION 67.</head>
        </head>
      </chunk>
      <chunk pos="437" start="1729880" end="1734372" type="number" n="68" id="elem.10.68" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D68">
        <head lang="en">
          <head>PROPOSITION 68.</head>
        </head>
      </chunk>
      <chunk pos="438" start="1734378" end="1736088" type="number" n="69" id="elem.10.69" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D69">
        <head lang="en">
          <head>PROPOSITION 69.</head>
        </head>
      </chunk>
      <chunk pos="439" start="1736114" end="1738191" type="number" n="70" id="elem.10.70" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D70">
        <head lang="en">
          <head>PROPOSITION 70.</head>
        </head>
      </chunk>
      <chunk pos="440" start="1738197" end="1745870" type="number" n="71" id="elem.10.71" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D71">
        <head lang="en">
          <head>PROPOSITION 71.</head>
        </head>
      </chunk>
      <chunk pos="441" start="1745876" end="1750502" type="number" n="72" id="elem.10.72" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D72">
        <head lang="en">
          <head>PROPOSITION 72.</head>
        </head>
      </chunk>
      <chunk pos="442" start="1750508" end="1752878" type="number" n="73" id="elem.10.73" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D73">
        <head lang="en">
          <head>PROPOSITION 73.</head>
        </head>
      </chunk>
      <chunk pos="443" start="1752884" end="1755070" type="number" n="74" id="elem.10.74" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D74">
        <head lang="en">
          <head>PROPOSITION 74.</head>
        </head>
      </chunk>
      <chunk pos="444" start="1755096" end="1760404" type="number" n="75" id="elem.10.75" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D75">
        <head lang="en">
          <head>PROPOSITION 75</head>
        </head>
      </chunk>
      <chunk pos="445" start="1760410" end="1762263" type="number" n="76" id="elem.10.76" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D76">
        <head lang="en">
          <head>PROPOSITION 76</head>
        </head>
      </chunk>
      <chunk pos="446" start="1762269" end="1764132" type="number" n="77" id="elem.10.77" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D77">
        <head lang="en">
          <head>PROPOSITION 77</head>
        </head>
      </chunk>
      <chunk pos="447" start="1764138" end="1768256" type="number" n="78" id="elem.10.78" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D78">
        <head lang="en">
          <head>PROPOSITION 78</head>
        </head>
      </chunk>
      <chunk pos="448" start="1768262" end="1770915" type="number" n="79" id="elem.10.79" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D79">
        <head lang="en">
          <head>PROPOSITION 79</head>
        </head>
      </chunk>
      <chunk pos="449" start="1770921" end="1773753" type="number" n="80" id="elem.10.80" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D80">
        <head lang="en">
          <head>PROPOSITION 80.</head>
        </head>
      </chunk>
      <chunk pos="450" start="1773779" end="1780080" type="number" n="81" id="elem.10.81" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D81">
        <head lang="en">
          <head>PROPOSITION 81.</head>
        </head>
      </chunk>
      <chunk pos="451" start="1780086" end="1782358" type="number" n="82" id="elem.10.82" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D82">
        <head lang="en">
          <head>PROPOSITION 82.</head>
        </head>
      </chunk>
      <chunk pos="452" start="1782364" end="1784721" type="number" n="83" id="elem.10.83" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D83">
        <head lang="en">
          <head>PROPOSITION 83.</head>
        </head>
      </chunk>
      <chunk pos="453" start="1784727" end="1790106" type="number" n="84" id="elem.10.84" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+2%3Anumber%3D84">
        <head lang="en">
          <head>PROPOSITION 84.</head>
        </head>
      </chunk>
    </chunk>
    <chunk pos="23" start="1790123" end="1792238" type="type" n="Def 3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DDef+3">
      <head lang="en">
        <head>DEFINITIONS III.</head>
      </head>
      <chunk pos="454" start="1790224" end="1790671" type="number" n="1" id="elem.10.def.3.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DDef+3%3Anumber%3D1">
        <head lang="en">
          <head>1</head>
        </head>
      </chunk>
      <chunk pos="455" start="1790677" end="1791066" type="number" n="2" id="elem.10.def.3.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DDef+3%3Anumber%3D2">
        <head lang="en">
          <head>2</head>
        </head>
      </chunk>
      <chunk pos="456" start="1791072" end="1791464" type="number" n="3" id="elem.10.def.3.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DDef+3%3Anumber%3D3">
        <head lang="en">
          <head>3</head>
        </head>
      </chunk>
      <chunk pos="457" start="1791470" end="1791875" type="number" n="4" id="elem.10.def.3.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DDef+3%3Anumber%3D4">
        <head lang="en">
          <head>4</head>
        </head>
      </chunk>
      <chunk pos="458" start="1791881" end="1792059" type="number" n="5" id="elem.10.def.3.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DDef+3%3Anumber%3D5">
        <head lang="en">
          <head>5</head>
        </head>
      </chunk>
      <chunk pos="459" start="1792065" end="1792226" type="number" n="6" id="elem.10.def.3.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DDef+3%3Anumber%3D6">
        <head lang="en">
          <head>6</head>
        </head>
      </chunk>
    </chunk>
    <chunk pos="24" start="1792262" end="1953965" type="type" n="Prop 3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3">
      <head lang="en">
        <head>PROPOSITIONS 85?115.</head>
      </head>
      <chunk pos="460" start="1792370" end="1796110" type="number" n="85" id="elem.10.85" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D85">
        <head lang="en">
          <head>PROPOSITION 85.</head>
        </head>
      </chunk>
      <chunk pos="461" start="1796136" end="1799659" type="number" n="86" id="elem.10.86" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D86">
        <head lang="en">
          <head>PROPOSITION 86.</head>
        </head>
      </chunk>
      <chunk pos="462" start="1799665" end="1805674" type="number" n="87" id="elem.10.87" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D87">
        <head lang="en">
          <head>PROPOSITION 87.</head>
        </head>
      </chunk>
      <chunk pos="463" start="1805680" end="1809357" type="number" n="88" id="elem.10.88" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D88">
        <head lang="en">
          <head>PROPOSITION 88.</head>
        </head>
      </chunk>
      <chunk pos="464" start="1809363" end="1813048" type="number" n="89" id="elem.10.89" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D89">
        <head lang="en">
          <head>PROPOSITION 89.</head>
        </head>
      </chunk>
      <chunk pos="465" start="1813074" end="1819248" type="number" n="90" id="elem.10.90" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D90">
        <head lang="en">
          <head>PROPOSITION 90.</head>
        </head>
      </chunk>
      <chunk pos="466" start="1819254" end="1827220" type="number" n="91" id="elem.10.91" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D91">
        <head lang="en">
          <head>PROPOSITION 91.</head>
        </head>
      </chunk>
      <chunk pos="467" start="1827226" end="1834839" type="number" n="92" id="elem.10.92" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D92">
        <head lang="en">
          <head>PROPOSITION 92.</head>
        </head>
      </chunk>
      <chunk pos="468" start="1834865" end="1844337" type="number" n="93" id="elem.10.93" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D93">
        <head lang="en">
          <head>PROPOSITION 93.</head>
        </head>
      </chunk>
      <chunk pos="469" start="1844363" end="1851195" type="number" n="94" id="elem.10.94" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D94">
        <head lang="en">
          <head>PROPOSITION 94.</head>
        </head>
      </chunk>
      <chunk pos="470" start="1851201" end="1855692" type="number" n="95" id="elem.10.95" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D95">
        <head lang="en">
          <head>PROPOSITION 95.</head>
        </head>
      </chunk>
      <chunk pos="471" start="1855718" end="1861571" type="number" n="96" id="elem.10.96" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D96">
        <head lang="en">
          <head>PROPOSITION 96.</head>
        </head>
      </chunk>
      <chunk pos="472" start="1861597" end="1868571" type="number" n="97" id="elem.10.97" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D97">
        <head lang="en">
          <head>PROPOSITION 97.</head>
        </head>
      </chunk>
      <chunk pos="473" start="1868577" end="1874944" type="number" n="98" id="elem.10.98" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D98">
        <head lang="en">
          <head>PROPOSITION 98.</head>
        </head>
      </chunk>
      <chunk pos="474" start="1874970" end="1882572" type="number" n="99" id="elem.10.99" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D99">
        <head lang="en">
          <head>PROPOSITION 99.</head>
        </head>
      </chunk>
      <chunk pos="475" start="1882578" end="1890286" type="number" n="100" id="elem.10.100" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D100">
        <head lang="en">
          <head>PROPOSITION 100.</head>
        </head>
      </chunk>
      <chunk pos="476" start="1890292" end="1896035" type="number" n="101" id="elem.10.101" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D101">
        <head lang="en">
          <head>PROPOSITION 101.</head>
        </head>
      </chunk>
      <chunk pos="477" start="1896061" end="1902519" type="number" n="102" id="elem.10.102" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D102">
        <head lang="en">
          <head>PROPOSITION 102.</head>
        </head>
      </chunk>
      <chunk pos="478" start="1902525" end="1907003" type="number" n="103" id="elem.10.103" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D103">
        <head lang="en">
          <head>PROPOSITION 103.</head>
        </head>
      </chunk>
      <chunk pos="479" start="1907009" end="1910462" type="number" n="104" id="elem.10.104" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D104">
        <head lang="en">
          <head>PROPOSITION 104.</head>
        </head>
      </chunk>
      <chunk pos="480" start="1910468" end="1913964" type="number" n="105" id="elem.10.105" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D105">
        <head lang="en">
          <head>PROPOSITION 105.</head>
        </head>
      </chunk>
      <chunk pos="481" start="1913970" end="1915987" type="number" n="106" id="elem.10.106" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D106">
        <head lang="en">
          <head>PROPOSITION 106.</head>
        </head>
      </chunk>
      <chunk pos="482" start="1915993" end="1918023" type="number" n="107" id="elem.10.107" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D107">
        <head lang="en">
          <head>PROPOSITION 107.</head>
        </head>
      </chunk>
      <chunk pos="483" start="1918049" end="1921769" type="number" n="108" id="elem.10.108" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D108">
        <head lang="en">
          <head>PROPOSITION 108.</head>
        </head>
      </chunk>
      <chunk pos="484" start="1921775" end="1924817" type="number" n="109" id="elem.10.109" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D109">
        <head lang="en">
          <head>PROPOSITION 109.</head>
        </head>
      </chunk>
      <chunk pos="485" start="1924823" end="1928457" type="number" n="110" id="elem.10.110" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D110">
        <head lang="en">
          <head>PROPOSITION 110.</head>
        </head>
      </chunk>
      <chunk pos="486" start="1928463" end="1931620" type="number" n="111" id="elem.10.111" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D111">
        <head lang="en">
          <head>PROPOSITION 111.</head>
        </head>
      </chunk>
      <chunk pos="487" start="1931626" end="1939401" type="number" n="112" id="elem.10.112" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D112">
        <head lang="en">
          <head>PROPOSITION 112.</head>
        </head>
      </chunk>
      <chunk pos="488" start="1939407" end="1947778" type="number" n="113" id="elem.10.113" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D113">
        <head lang="en">
          <head>PROPOSITION 113.</head>
        </head>
      </chunk>
      <chunk pos="489" start="1947784" end="1951928" type="number" n="114" id="elem.10.114" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D114">
        <head lang="en">
          <head>PROPOSITION 114.</head>
        </head>
      </chunk>
      <chunk pos="490" start="1951934" end="1953953" type="number" n="115" id="elem.10.115" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D10%3Atype%3DProp+3%3Anumber%3D115">
        <head lang="en">
          <head>PROPOSITION 115.</head>
        </head>
      </chunk>
    </chunk>
  </chunk>
  <chunk pos="11" start="1953980" end="2112030" type="book" n="11" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11">
    <head lang="en">
      <head>BOOK XI.</head>
    </head>
    <chunk pos="25" start="1954087" end="1962211" type="type" n="Def" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef">
      <head lang="en">
        <head>DEFINITIONS.</head>
      </head>
      <chunk pos="491" start="1954182" end="1954369" type="number" n="1" id="elem.11.def.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D1">
        <head lang="en">
          <head>1</head>
        </head>
      </chunk>
      <chunk pos="492" start="1954375" end="1954525" type="number" n="2" id="elem.11.def.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D2">
        <head lang="en">
          <head>2</head>
        </head>
      </chunk>
      <chunk pos="493" start="1954531" end="1954823" type="number" n="3" id="elem.11.def.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D3">
        <head lang="en">
          <head>3</head>
        </head>
      </chunk>
      <chunk pos="494" start="1954829" end="1955149" type="number" n="4" id="elem.11.def.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D4">
        <head lang="en">
          <head>4</head>
        </head>
      </chunk>
      <chunk pos="495" start="1955155" end="1955658" type="number" n="5" id="elem.11.def.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D5">
        <head lang="en">
          <head>5</head>
        </head>
      </chunk>
      <chunk pos="496" start="1955664" end="1955976" type="number" n="6" id="elem.11.def.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D6">
        <head lang="en">
          <head>6</head>
        </head>
      </chunk>
      <chunk pos="497" start="1955982" end="1956259" type="number" n="7" id="elem.11.def.7" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D7">
        <head lang="en">
          <head>7</head>
        </head>
      </chunk>
      <chunk pos="498" start="1956265" end="1956443" type="number" n="8" id="elem.11.def.8" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D8">
        <head lang="en">
          <head>8</head>
        </head>
      </chunk>
      <chunk pos="499" start="1956469" end="1956682" type="number" n="9" id="elem.11.def.9" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D9">
        <head lang="en">
          <head>9</head>
        </head>
      </chunk>
      <chunk pos="500" start="1956688" end="1956931" type="number" n="10" id="elem.11.def.10" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D10">
        <head lang="en">
          <head>10</head>
        </head>
      </chunk>
      <chunk pos="501" start="1956937" end="1957403" type="number" n="11" id="elem.11.def.11" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D11">
        <head lang="en">
          <head>11</head>
        </head>
      </chunk>
      <chunk pos="502" start="1957409" end="1957645" type="number" n="12" id="elem.11.def.12" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D12">
        <head lang="en">
          <head>12</head>
        </head>
      </chunk>
      <chunk pos="503" start="1957651" end="1957948" type="number" n="13" id="elem.11.def.13" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D13">
        <head lang="en">
          <head>13</head>
        </head>
      </chunk>
      <chunk pos="504" start="1957954" end="1958287" type="number" n="14" id="elem.11.def.14" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D14">
        <head lang="en">
          <head>14</head>
        </head>
      </chunk>
      <chunk pos="505" start="1958293" end="1958535" type="number" n="15" id="elem.11.def.15" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D15">
        <head lang="en">
          <head>15</head>
        </head>
      </chunk>
      <chunk pos="506" start="1958541" end="1958741" type="number" n="16" id="elem.11.def.16" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D16">
        <head lang="en">
          <head>16</head>
        </head>
      </chunk>
      <chunk pos="507" start="1958747" end="1959018" type="number" n="17" id="elem.11.def.17" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D17">
        <head lang="en">
          <head>17</head>
        </head>
      </chunk>
      <chunk pos="508" start="1959024" end="1959682" type="number" n="18" id="elem.11.def.18" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D18">
        <head lang="en">
          <head>18</head>
        </head>
      </chunk>
      <chunk pos="509" start="1959688" end="1959926" type="number" n="19" id="elem.11.def.19" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D19">
        <head lang="en">
          <head>19</head>
        </head>
      </chunk>
      <chunk pos="510" start="1959952" end="1960170" type="number" n="20" id="elem.11.def.20" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D20">
        <head lang="en">
          <head>20</head>
        </head>
      </chunk>
      <chunk pos="511" start="1960176" end="1960556" type="number" n="21" id="elem.11.def.21" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D21">
        <head lang="en">
          <head>21</head>
        </head>
      </chunk>
      <chunk pos="512" start="1960562" end="1960809" type="number" n="22" id="elem.11.def.22" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D22">
        <head lang="en">
          <head>22</head>
        </head>
      </chunk>
      <chunk pos="513" start="1960815" end="1961057" type="number" n="23" id="elem.11.def.23" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D23">
        <head lang="en">
          <head>23</head>
        </head>
      </chunk>
      <chunk pos="514" start="1961063" end="1961304" type="number" n="24" id="elem.11.def.24" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D24">
        <head lang="en">
          <head>24</head>
        </head>
      </chunk>
      <chunk pos="515" start="1961310" end="1961503" type="number" n="25" id="elem.11.def.25" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D25">
        <head lang="en">
          <head>25</head>
        </head>
      </chunk>
      <chunk pos="516" start="1961509" end="1961729" type="number" n="26" id="elem.11.def.26" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D26">
        <head lang="en">
          <head>26</head>
        </head>
      </chunk>
      <chunk pos="517" start="1961735" end="1961957" type="number" n="27" id="elem.11.def.27" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D27">
        <head lang="en">
          <head>27</head>
        </head>
      </chunk>
      <chunk pos="518" start="1961963" end="1962199" type="number" n="28" id="elem.11.def.28" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DDef%3Anumber%3D28">
        <head lang="en">
          <head>28</head>
        </head>
      </chunk>
    </chunk>
    <chunk pos="26" start="1962273" end="2112019" type="type" n="Prop" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp">
      <head lang="en">
        <head>BOOK XI. PROPOSITIONS.</head>
      </head>
      <chunk pos="519" start="1962379" end="1963960" type="number" n="1" id="elem.11.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D1">
        <head lang="en">
          <head>PROPOSITION 1.</head>
        </head>
      </chunk>
      <chunk pos="520" start="1963966" end="1966264" type="number" n="2" id="elem.11.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D2">
        <head lang="en">
          <head>PROPOSITION 2.</head>
        </head>
      </chunk>
      <chunk pos="521" start="1966270" end="1967610" type="number" n="3" id="elem.11.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D3">
        <head lang="en">
          <head>PROPOSITION 3.</head>
        </head>
      </chunk>
      <chunk pos="522" start="1967616" end="1973354" type="number" n="4" id="elem.11.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D4">
        <head lang="en">
          <head>PROPOSITION 4.</head>
        </head>
      </chunk>
      <chunk pos="523" start="1973360" end="1976296" type="number" n="5" id="elem.11.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D5">
        <head lang="en">
          <head>PROPOSITION 5.</head>
        </head>
      </chunk>
      <chunk pos="524" start="1976302" end="1980135" type="number" n="6" id="elem.11.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D6">
        <head lang="en">
          <head>PROPOSITION 6.</head>
        </head>
      </chunk>
      <chunk pos="525" start="1980141" end="1981731" type="number" n="7" id="elem.11.7" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D7">
        <head lang="en">
          <head>PROPOSITION 7.</head>
        </head>
      </chunk>
      <chunk pos="526" start="1981737" end="1986831" type="number" n="8" id="elem.11.8" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D8">
        <head lang="en">
          <head>PROPOSITION 8.</head>
        </head>
      </chunk>
      <chunk pos="527" start="1986837" end="1988981" type="number" n="9" id="elem.11.9" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D9">
        <head lang="en">
          <head>PROPOSITION 9.</head>
        </head>
      </chunk>
      <chunk pos="528" start="1989007" end="1991432" type="number" n="10" id="elem.11.10" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D10">
        <head lang="en">
          <head>PROPOSITION 10.</head>
        </head>
      </chunk>
      <chunk pos="529" start="1991438" end="1995184" type="number" n="11" id="elem.11.11" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D11">
        <head lang="en">
          <head>PROPOSITION 11.</head>
        </head>
      </chunk>
      <chunk pos="530" start="1995190" end="1996599" type="number" n="12" id="elem.11.12" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D12">
        <head lang="en">
          <head>PROPOSITION 12.</head>
        </head>
      </chunk>
      <chunk pos="531" start="1996625" end="1998242" type="number" n="13" id="elem.11.13" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D13">
        <head lang="en">
          <head>PROPOSITION 13.</head>
        </head>
      </chunk>
      <chunk pos="532" start="1998268" end="2000179" type="number" n="14" id="elem.11.14" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D14">
        <head lang="en">
          <head>PROPOSITION 14.</head>
        </head>
      </chunk>
      <chunk pos="533" start="2000205" end="2003652" type="number" n="15" id="elem.11.15" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D15">
        <head lang="en">
          <head>PROPOSITION 15.</head>
        </head>
      </chunk>
      <chunk pos="534" start="2003678" end="2006022" type="number" n="16" id="elem.11.16" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D16">
        <head lang="en">
          <head>PROPOSITION 16.</head>
        </head>
      </chunk>
      <chunk pos="535" start="2006028" end="2008764" type="number" n="17" id="elem.11.17" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D17">
        <head lang="en">
          <head>PROPOSITION 17.</head>
        </head>
      </chunk>
      <chunk pos="536" start="2008770" end="2011416" type="number" n="18" id="elem.11.18" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D18">
        <head lang="en">
          <head>PROPOSITION 18.</head>
        </head>
      </chunk>
      <chunk pos="537" start="2011422" end="2013403" type="number" n="19" id="elem.11.19" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D19">
        <head lang="en">
          <head>PROPOSITION 19.</head>
        </head>
      </chunk>
      <chunk pos="538" start="2013409" end="2016567" type="number" n="20" id="elem.11.20" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D20">
        <head lang="en">
          <head>PROPOSITION 20.</head>
        </head>
      </chunk>
      <chunk pos="539" start="2016573" end="2019793" type="number" n="21" id="elem.11.21" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D21">
        <head lang="en">
          <head>PROPOSITION 21.</head>
        </head>
      </chunk>
      <chunk pos="540" start="2019799" end="2024448" type="number" n="22" id="elem.11.22" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D22">
        <head lang="en">
          <head>PROPOSITION 22.</head>
        </head>
      </chunk>
      <chunk pos="541" start="2024454" end="2036726" type="number" n="23" id="elem.11.23" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D23">
        <head lang="en">
          <head>PROPOSITION 23.</head>
        </head>
      </chunk>
      <chunk pos="542" start="2036732" end="2040109" type="number" n="24" id="elem.11.24" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D24">
        <head lang="en">
          <head>PROPOSITION 24.</head>
        </head>
      </chunk>
      <chunk pos="543" start="2040135" end="2044849" type="number" n="25" id="elem.11.25" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D25">
        <head lang="en">
          <head>PROPOSITION 25.</head>
        </head>
      </chunk>
      <chunk pos="544" start="2044855" end="2049651" type="number" n="26" id="elem.11.26" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D26">
        <head lang="en">
          <head>PROPOSITION 26.</head>
        </head>
      </chunk>
      <chunk pos="545" start="2049657" end="2052819" type="number" n="27" id="elem.11.27" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D27">
        <head lang="en">
          <head>PROPOSITION 27.</head>
        </head>
      </chunk>
      <chunk pos="546" start="2052825" end="2054573" type="number" n="28" id="elem.11.28" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D28">
        <head lang="en">
          <head>PROPOSITION 28.</head>
        </head>
      </chunk>
      <chunk pos="547" start="2054579" end="2057503" type="number" n="29" id="elem.11.29" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D29">
        <head lang="en">
          <head>PROPOSITION 29.</head>
        </head>
      </chunk>
      <chunk pos="548" start="2057509" end="2060543" type="number" n="30" id="elem.11.30" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D30">
        <head lang="en">
          <head>PROPOSITION 30.</head>
        </head>
      </chunk>
      <chunk pos="549" start="2060549" end="2069189" type="number" n="31" id="elem.11.31" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D31">
        <head lang="en">
          <head>PROPOSITION 31.</head>
        </head>
      </chunk>
      <chunk pos="550" start="2069195" end="2071230" type="number" n="32" id="elem.11.32" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D32">
        <head lang="en">
          <head>PROPOSITION 32.</head>
        </head>
      </chunk>
      <chunk pos="551" start="2071236" end="2077742" type="number" n="33" id="elem.11.33" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D33">
        <head lang="en">
          <head>PROPOSITION 33.</head>
        </head>
      </chunk>
      <chunk pos="552" start="2077748" end="2090165" type="number" n="34" id="elem.11.34" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D34">
        <head lang="en">
          <head>PROPOSITION 34.</head>
        </head>
      </chunk>
      <chunk pos="553" start="2090191" end="2098926" type="number" n="35" id="elem.11.35" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D35">
        <head lang="en">
          <head>PROPOSITION 35.</head>
        </head>
      </chunk>
      <chunk pos="554" start="2098932" end="2102949" type="number" n="36" id="elem.11.36" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D36">
        <head lang="en">
          <head>PROPOSITION 36.</head>
        </head>
      </chunk>
      <chunk pos="555" start="2102955" end="2106062" type="number" n="37" id="elem.11.37" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D37">
        <head lang="en">
          <head>PROPOSITION 37.</head>
        </head>
      </chunk>
      <chunk pos="556" start="2106068" end="2110022" type="number" n="38" id="elem.11.38" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D38">
        <head lang="en">
          <head>PROPOSITION 38.</head>
        </head>
      </chunk>
      <chunk pos="557" start="2110028" end="2112007" type="number" n="39" id="elem.11.39" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D11%3Atype%3DProp%3Anumber%3D39">
        <head lang="en">
          <head>PROPOSITION 39.</head>
        </head>
      </chunk>
    </chunk>
  </chunk>
  <chunk pos="12" start="2112034" end="2253838" type="book" n="12" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12">
    <head lang="en">
      <head>BOOK XII.</head>
    </head>
    <chunk pos="27" start="2126807" end="2253827" type="type" n="Prop" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12%3Atype%3DProp">
      <head lang="en">
        <head>BOOK XII. PROPOSITIONS.</head>
      </head>
      <chunk pos="558" start="2126914" end="2130287" type="number" n="1" id="elem.12.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12%3Atype%3DProp%3Anumber%3D1">
        <head lang="en">
          <head>PROPOSITION 1.</head>
        </head>
      </chunk>
      <chunk pos="559" start="2130313" end="2139640" type="number" n="2" id="elem.12.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12%3Atype%3DProp%3Anumber%3D2">
        <head lang="en">
          <head>PROPOSITION 2.</head>
        </head>
      </chunk>
      <chunk pos="560" start="2139646" end="2149598" type="number" n="3" id="elem.12.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12%3Atype%3DProp%3Anumber%3D3">
        <head lang="en">
          <head>PROPOSITION 3.</head>
        </head>
      </chunk>
      <chunk pos="561" start="2149604" end="2157976" type="number" n="4" id="elem.12.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12%3Atype%3DProp%3Anumber%3D4">
        <head lang="en">
          <head>PROPOSITION 4.</head>
        </head>
      </chunk>
      <chunk pos="562" start="2157982" end="2163310" type="number" n="5" id="elem.12.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12%3Atype%3DProp%3Anumber%3D5">
        <head lang="en">
          <head>PROPOSITION 5.</head>
        </head>
      </chunk>
      <chunk pos="563" start="2163336" end="2167229" type="number" n="6" id="elem.12.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12%3Atype%3DProp%3Anumber%3D6">
        <head lang="en">
          <head>PROPOSITION 6.</head>
        </head>
      </chunk>
      <chunk pos="564" start="2167255" end="2171518" type="number" n="7" id="elem.12.7" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12%3Atype%3DProp%3Anumber%3D7">
        <head lang="en">
          <head>PROPOSITION 7.</head>
        </head>
      </chunk>
      <chunk pos="565" start="2171524" end="2176644" type="number" n="8" id="elem.12.8" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12%3Atype%3DProp%3Anumber%3D8">
        <head lang="en">
          <head>PROPOSITION 8.</head>
        </head>
      </chunk>
      <chunk pos="566" start="2176670" end="2181895" type="number" n="9" id="elem.12.9" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12%3Atype%3DProp%3Anumber%3D9">
        <head lang="en">
          <head>PROPOSITION 9.</head>
        </head>
      </chunk>
      <chunk pos="567" start="2181901" end="2192236" type="number" n="10" id="elem.12.10" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12%3Atype%3DProp%3Anumber%3D10">
        <head lang="en">
          <head>PROPOSITION 10.</head>
        </head>
      </chunk>
      <chunk pos="568" start="2192262" end="2200778" type="number" n="11" id="elem.12.11" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12%3Atype%3DProp%3Anumber%3D11">
        <head lang="en">
          <head>PROPOSITION 11.</head>
        </head>
      </chunk>
      <chunk pos="569" start="2200804" end="2216542" type="number" n="12" id="elem.12.12" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12%3Atype%3DProp%3Anumber%3D12">
        <head lang="en">
          <head>PROPOSITION 12.</head>
        </head>
      </chunk>
      <chunk pos="570" start="2216548" end="2220931" type="number" n="13" id="elem.12.13" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12%3Atype%3DProp%3Anumber%3D13">
        <head lang="en">
          <head>PROPOSITION 13.</head>
        </head>
      </chunk>
      <chunk pos="571" start="2220937" end="2223411" type="number" n="14" id="elem.12.14" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12%3Atype%3DProp%3Anumber%3D14">
        <head lang="en">
          <head>PROPOSITION 14.</head>
        </head>
      </chunk>
      <chunk pos="572" start="2223437" end="2229434" type="number" n="15" id="elem.12.15" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12%3Atype%3DProp%3Anumber%3D15">
        <head lang="en">
          <head>PROPOSITION 15.</head>
        </head>
      </chunk>
      <chunk pos="573" start="2229440" end="2232035" type="number" n="16" id="elem.12.16" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12%3Atype%3DProp%3Anumber%3D16">
        <head lang="en">
          <head>PROPOSITION 16.</head>
        </head>
      </chunk>
      <chunk pos="574" start="2232061" end="2248810" type="number" n="17" id="elem.12.17" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12%3Atype%3DProp%3Anumber%3D17">
        <head lang="en">
          <head>PROPOSITION 17.</head>
        </head>
      </chunk>
      <chunk pos="575" start="2248816" end="2253815" type="number" n="18" id="elem.12.18" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D12%3Atype%3DProp%3Anumber%3D18">
        <head lang="en">
          <head>PROPOSITION 18.</head>
        </head>
      </chunk>
    </chunk>
  </chunk>
  <chunk pos="13" start="2253842" end="2376508" type="book" n="13" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13">
    <head lang="en">
      <head>BOOK XIII.</head>
    </head>
    <chunk pos="28" start="2258512" end="2376497" type="type" n="Prop" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13%3Atype%3DProp">
      <head lang="en">
        <head>BOOK XIII. PROPOSITIONS.</head>
      </head>
      <chunk pos="576" start="2258620" end="2262105" type="number" n="1" id="elem.13.1" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13%3Atype%3DProp%3Anumber%3D1">
        <head lang="en">
          <head>PROPOSITION 1.</head>
        </head>
      </chunk>
      <chunk pos="577" start="2262111" end="2266835" type="number" n="2" id="elem.13.2" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13%3Atype%3DProp%3Anumber%3D2">
        <head lang="en">
          <head>PROPOSITION 2.</head>
        </head>
      </chunk>
      <chunk pos="578" start="2266841" end="2269995" type="number" n="3" id="elem.13.3" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13%3Atype%3DProp%3Anumber%3D3">
        <head lang="en">
          <head>PROPOSITION 3.</head>
        </head>
      </chunk>
      <chunk pos="579" start="2270001" end="2272822" type="number" n="4" id="elem.13.4" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13%3Atype%3DProp%3Anumber%3D4">
        <head lang="en">
          <head>PROPOSITION 4.</head>
        </head>
      </chunk>
      <chunk pos="580" start="2272828" end="2275755" type="number" n="5" id="elem.13.5" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13%3Atype%3DProp%3Anumber%3D5">
        <head lang="en">
          <head>PROPOSITION 5.</head>
        </head>
      </chunk>
      <chunk pos="581" start="2275761" end="2279060" type="number" n="6" id="elem.13.6" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13%3Atype%3DProp%3Anumber%3D6">
        <head lang="en">
          <head>PROPOSITION 6.</head>
        </head>
      </chunk>
      <chunk pos="582" start="2279066" end="2283987" type="number" n="7" id="elem.13.7" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13%3Atype%3DProp%3Anumber%3D7">
        <head lang="en">
          <head>PROPOSITION 7.</head>
        </head>
      </chunk>
      <chunk pos="583" start="2283993" end="2288417" type="number" n="8" id="elem.13.8" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13%3Atype%3DProp%3Anumber%3D8">
        <head lang="en">
          <head>PROPOSITION 8.</head>
        </head>
      </chunk>
      <chunk pos="584" start="2288423" end="2292595" type="number" n="9" id="elem.13.9" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13%3Atype%3DProp%3Anumber%3D9">
        <head lang="en">
          <head>PROPOSITION 9.</head>
        </head>
      </chunk>
      <chunk pos="585" start="2292601" end="2299811" type="number" n="10" id="elem.13.10" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13%3Atype%3DProp%3Anumber%3D10">
        <head lang="en">
          <head>PROPOSITION 10.</head>
        </head>
      </chunk>
      <chunk pos="586" start="2299817" end="2309460" type="number" n="11" id="elem.13.11" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13%3Atype%3DProp%3Anumber%3D11">
        <head lang="en">
          <head>PROPOSITION 11.</head>
        </head>
      </chunk>
      <chunk pos="587" start="2309466" end="2311892" type="number" n="12" id="elem.13.12" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13%3Atype%3DProp%3Anumber%3D12">
        <head lang="en">
          <head>PROPOSITION 12.</head>
        </head>
      </chunk>
      <chunk pos="588" start="2311898" end="2321984" type="number" n="13" id="elem.13.13" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13%3Atype%3DProp%3Anumber%3D13">
        <head lang="en">
          <head>PROPOSITION 13.</head>
        </head>
      </chunk>
      <chunk pos="589" start="2322010" end="2327929" type="number" n="14" id="elem.13.14" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13%3Atype%3DProp%3Anumber%3D14">
        <head lang="en">
          <head>PROPOSITION 14.</head>
        </head>
      </chunk>
      <chunk pos="590" start="2327955" end="2332934" type="number" n="15" id="elem.13.15" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13%3Atype%3DProp%3Anumber%3D15">
        <head lang="en">
          <head>PROPOSITION 15.</head>
        </head>
      </chunk>
      <chunk pos="591" start="2332960" end="2346988" type="number" n="16" id="elem.13.16" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13%3Atype%3DProp%3Anumber%3D16">
        <head lang="en">
          <head>PROPOSITION 16.</head>
        </head>
      </chunk>
      <chunk pos="592" start="2347014" end="2361252" type="number" n="17" id="elem.13.17" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13%3Atype%3DProp%3Anumber%3D17">
        <head lang="en">
          <head>PROPOSITION 17.</head>
        </head>
      </chunk>
      <chunk pos="593" start="2361258" end="2376485" type="number" n="18" id="elem.13.18" ref="Perseus%3Atext%3A1999.01.0086%3Abook%3D13%3Atype%3DProp%3Anumber%3D18">
        <head lang="en">
          <head>PROPOSITION 18.</head>
        </head>
      </chunk>
    </chunk>
  </chunk>
</contents>