83 lines
9.6 KiB
HTML
83 lines
9.6 KiB
HTML
<html><head>
|
||
<meta http-equiv="content-type" content="text/html; charset=windows-1252">
|
||
<title>Euclid's Elements, Book I, Definition 2</title>
|
||
<link rel="stylesheet" type="text/css" href="joyceDefI2_files/elements.css">
|
||
<script src="joyceDefI2_files/elements.js"></script>
|
||
</head>
|
||
<body>
|
||
|
||
<div id="header"><a class="hover" href="https://mathcs.clarku.edu/%7Edjoyce/java/elements/elements.html"><h1>Euclid's Elements</h1></a><a class="hover" href="https://mathcs.clarku.edu/%7Edjoyce/java/elements/bookI/bookI.html"><h1>Book I</h1></a></div>
|
||
<script>loadHeader();</script>
|
||
|
||
<div class="theorem">
|
||
<h1>Definition 2</h1>
|
||
|
||
<div class="statement">A <i>line</i> is breadthless length.</div>
|
||
</div>
|
||
|
||
<a name="guide"><h2>Guide</h2></a>
|
||
|
||
“Line” is the second primitive term in the <i>Elements.</i> The
|
||
description, “breadthless length,” says that a line will have one
|
||
dimension, length, but it won’t have breadth. The terms length and
|
||
breadth are not defined in the <i>Elements.</i> In <a href="https://mathcs.clarku.edu/%7Edjoyce/java/elements/bookI/defI5.html">I.Def.5</a> a surface is defined with the two dimensions length and breadth, and in <a href="https://mathcs.clarku.edu/%7Edjoyce/java/elements/bookXI/defXI1.html">XI.Def.1</a> a solid is defined with the three dimensions length, breadth, and depth.
|
||
|
||
<table cellspacing="5">
|
||
<tbody><tr><td>
|
||
<p>One cannot tell from this definition what kind of line is meant by
|
||
line, but later a straight line defined to be a special kind of line.
|
||
One can conclude, then, that lines need not be straight. Perhaps
|
||
“curve” would be a better translation than line since Euclid meant what
|
||
is commonly called a curve in modern English, where a curve may or may
|
||
not be straight.
|
||
|
||
</p><p>Also, from the next definition, it is apparent that Euclid’s
|
||
lines may have ends, so they are line segments or curve segments. But
|
||
they need not have ends in all cases since the entire circumference of a
|
||
circle is an example of a line. Indeed, lines need not be finite in
|
||
all cases; there are a few instances in the <i>Elements</i> where a line is not bounded, and that is usually indicated by the language. See, for example, proposition <a href="https://mathcs.clarku.edu/%7Edjoyce/java/elements/bookI/propI12.html">I.12</a>.
|
||
</p></td>
|
||
<td>
|
||
<div id="ap1">
|
||
<applet code="Geometry" archive="Geometry.zip" width="320" height="200">
|
||
<img src="joyceDefI2_files/defI2.gif" alt="java applet or image">
|
||
<param name="background" value="35,19,100">
|
||
<param name="title" value="I.39">
|
||
<param name="e[1]" value="A;point;free;40,170;0">
|
||
<param name="e[2]" value="B;point;free;110,100;0">
|
||
<param name="e[3]" value="C;point;free;190,100;0">
|
||
<param name="e[4]" value="D;point;free;280,50;0">
|
||
<param name="e[5]" value="BC;line;connect;B,C">
|
||
<param name="e[6]" value="E';point;perpendicular;B,C;0;0">
|
||
<param name="e[7]" value="M;point;midpoint;A,B;0;0">
|
||
<param name="e[8]" value="E'';point;perpendicular;M,B;0;0">
|
||
<param name="e[9]" value="E;point;intersection;B,E',M,E'';0;0">
|
||
<param name="e[10]" value="AMB;sector;sector;E,B,A;0;0;black;0">
|
||
<param name="e[11]" value="F';point;perpendicular;C,B;0;0">
|
||
<param name="e[12]" value="N;point;midpoint;C,D;0;0">
|
||
<param name="e[13]" value="F'';point;perpendicular;N,C;0;0">
|
||
<param name="e[14]" value="F;point;intersection;C,F',N,F'';0;0">
|
||
<param name="e[15]" value="CND;sector;sector;F,C,D;0;0;black;0">
|
||
</applet>
|
||
</div>
|
||
<script>
|
||
if (!navigator.javaEnabled()) document.getElementById('ap1').innerHTML =
|
||
'<img src="defI2.gif" hspace=10 vspace=10 alt="I.Def.2">';
|
||
</script>
|
||
</td></tr>
|
||
</tbody></table>
|
||
|
||
|
||
<p>One piece of terminology that Euclid did not mention explicitly in a
|
||
definition is a phrase to indicate when a line passes through a point.
|
||
That would be a primitive relation that could hold between a line and a
|
||
point. Postulates would be included as well to give meaning to the
|
||
phrase as they are in modern treatments of elementary geometry.
|
||
|
||
</p><div id="footer"><br><p></p><hr size="6"><p></p><center><table><tbody><tr valign="top"><td>Next: <a href="https://mathcs.clarku.edu/%7Edjoyce/java/elements/bookI/defI3.html">I.Def.3</a> </td><td><form name="PropForm"><select name="select1" size="1" onchange="if (options[selectedIndex].value){ location = options[selectedIndex].value }"><option value="" selected="selected">Select from Book I</option><option value="bookI.html">Book I intro</option><option value="defI1.html">I.Def.1</option><option value="defI2.html">I.Def.2</option><option value="defI3.html">I.Def.3</option><option value="defI4.html">I.Def.4</option><option value="defI5.html">I.Def.5</option><option value="defI6.html">I.Def.6</option><option value="defI7.html">I.Def.7</option><option value="defI8.html">I.Def.8</option><option value="defI9.html">I.Def.9</option><option value="defI10.html">I.Def.10</option><option value="defI11.html">I.Def.11-12</option><option value="defI13.html">I.Def.13-14</option><option value="defI15.html">I.Def.15-18</option><option value="defI19.html">I.Def.19</option><option value="defI20.html">I.Def.21</option><option value="defI22.html">I.Def.22</option><option value="defI23.html">I.Def.23</option><option value="cn.html">Common Notions</option><option value="post1.html">I.Post.1</option><option value="post2.html">I.Post.2</option><option value="post3.html">I.Post.3</option><option value="post4.html">I.Post.4</option><option value="post5.html">I.Post.5</option><option value="propI1.html">I.1</option><option value="propI2.html">I.2</option><option value="propI3.html">I.3</option><option value="propI4.html">I.4</option><option value="propI5.html">I.5</option><option value="propI6.html">I.6</option><option value="propI7.html">I.7</option><option value="propI8.html">I.8</option><option value="propI9.html">I.9</option><option value="propI10.html">I.10</option><option value="propI11.html">I.11</option><option value="propI12.html">I.12</option><option value="propI13.html">I.13</option><option value="propI14.html">I.14</option><option value="propI15.html">I.15</option><option value="propI16.html">I.16</option><option value="propI17.html">I.17</option><option value="propI18.html">I.18</option><option value="propI19.html">I.19</option><option value="propI20.html">I.20</option><option value="propI21.html">I.21</option><option value="propI22.html">I.22</option><option value="propI23.html">I.23</option><option value="propI24.html">I.24</option><option value="propI25.html">I.25</option><option value="propI26.html">I.26</option><option value="propI27.html">I.27</option><option value="propI28.html">I.28</option><option value="propI29.html">I.29</option><option value="propI30.html">I.30</option><option value="propI31.html">I.31</option><option value="propI32.html">I.32</option><option value="propI33.html">I.33</option><option value="propI34.html">I.34</option><option value="propI35.html">I.35</option><option value="propI36.html">I.36</option><option value="propI37.html">I.37</option><option value="propI38.html">I.38</option><option value="propI39.html">I.39</option><option value="propI40.html">I.40</option><option value="propI41.html">I.41</option><option value="propI42.html">I.42</option><option value="propI43.html">I.43</option><option value="propI44.html">I.44</option><option value="propI45.html">I.45</option><option value="propI46.html">I.46</option><option value="propI47.html">I.47</option><option value="propI48.html">I.48</option></select></form></td></tr><tr valign="top"><td>Previous: <a href="https://mathcs.clarku.edu/%7Edjoyce/java/elements/bookI/defI1.html">I.Def.1</a></td><td><form name="BookForm"><select name="select2" size="1" onchange="if (options[selectedIndex].value){ location = options[selectedIndex].value }"><option value="" selected="selected">Select book</option><option value="../bookI/bookI.html">Book I</option><option value="../bookII/bookII.html">Book II</option><option value="../bookIII/bookIII.html">Book III</option><option value="../bookIV/bookIV.html">Book IV</option><option value="../bookV/bookV.html">Book V</option><option value="../bookVI/bookVI.html">Book VI</option><option value="../bookVII/bookVII.html">Book VII</option><option value="../bookVIII/bookVIII.html">Book VIII</option><option value="../bookIX/bookIX.html">Book IX</option><option value="../bookX/bookX.html">Book X</option><option value="../bookXI/bookXI.html">Book XI</option><option value="../bookXII/bookXII.html">Book XII</option><option value="../bookXIII/bookXIII.html">Book XIII</option></select></form></td></tr><tr valign="top"><td><a href="https://mathcs.clarku.edu/%7Edjoyce/java/elements/bookI/bookI.html">Book I</a></td><td><form name="TopicForm"><select name="select3" size="1" onchange="if (options[selectedIndex].value){ location = options[selectedIndex].value }"><option value="" selected="selected">Select topic</option><option value="../elements.html">Introduction</option><option value="../toc.html">Table of Contents</option><option value="../usingApplet.html">Geometry applet</option><option value="../aboutText.html">About the text</option><option value="../Euclid.html">Euclid</option><option value="../web.html">Web references</option><option value="../trip.html">A quick trip</option><option value="../subjindex.html">Subject index</option></select></form></td></tr></tbody></table></center><hr size="6"><p><font size="-1"><a href="https://mathcs.clarku.edu/%7Edjoyce/java/elements/copyright.html">©1996</a><br><a href="http://aleph0.clarku.edu/%7Edjoyce/">David E. Joyce</a><br><img src="joyceDefI2_files/dj.gif"><br><a href="http://www.clarku.edu/departments/mathcs/">Department of Mathematics and Computer Science</a><br><a href="http://www.clarku.edu/">Clark University</a><br>Worcester, MA 01610</font></p></div>
|
||
<script>
|
||
loadFooter("1996");
|
||
</script>
|
||
|
||
</body></html>
|