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				IMSS - V.2. The "pyramidal" law</title>
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<td align="center"><img src="http://vitruvio.imss.fi.it/foto/mdl/V_2rs.jpg" name="ingrandimento" alt="V.2" border="1" class="leoimggalleria"></td>
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<td width="240" colspan="3" align="left"><span><span id="Filmato"></span></span><br><span class="leotitoloimmag"><span id="testoTitolo">V.2</span></span><span class="leodidascalia"><span id="testoDida"><br>The "pyramidal" law</span></span></td>
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<td><span class="parent-liv1">[</span><span class="numeri-liv2">V.2</span>  <span class="titolo-liv2">The "pyramidal" law</span><span class="parent-liv1">]</span></td>
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<p><campo tipocampo="T" nome="txtxml" style=""><div class="Section1" xmlns:w="http://schemas.microsoft.com/office/word/2003/wordml" xmlns:o="urn:schemas-microsoft-com:office:office" xmlns:v="urn:schemas-microsoft-com:vml" xmlns:WX="http://schemas.microsoft.com/office/word/2003/auxHint" xmlns:aml="http://schemas.microsoft.com/aml/2001/core" xmlns:w10="urn:schemas-microsoft-com:office:word">
<p class="Normal-P" style="text-align:justify;text-justify:inter-ideograph;"><span class="Normal-H">Augmentation and diminution in the natural "powers" take place in "pyramidal" &#8211; that is, strictly proportional &#8211; manner. For Leonardo the velocity at which a falling body drops increases in uniform manner, just as that of a weight thrown vertically upward decreases uniformly. In the same way, force augments with velocity and attenuates with distance. And weight and percussion increase in proportion to velocity. The pyramidal law represents for Leonardo a universal constant.</span></p>
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<pre><img src="images/spacer.gif" alt="0" width="47" height="60" align="absmiddle"><img src="http://vitruvio.imss.fi.it/foto/mdl/V_2r.gif" alt="V.2" name="img00" hspace="2" align="absmiddle" onclick="cambiaImg(0,'http://vitruvio.imss.fi.it/foto/mdl/V_2rs.jpg', aggiustaApici('The &quot;pyramidal&quot; law'), aggiustaApici('V.2'), '')" style="cursor:pointer; border-color:#98907d" onmouseover="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2r.gif', true)" onmouseout="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2r.gif', false)" border="1"><script>
				totIcone++;
				pathImm["img"+totIcone] = 'http://vitruvio.imss.fi.it/foto/mdl/V_2rs.jpg';
				didascalie[totIcone]='The "pyramidal" law';
				titolo[totIcone]='V.2';
				chiaveImm[totIcone]='13840';
				filmato[totIcone]='';
			</script></pre>
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<pre><img src="images/a.gif" alt="a" width="47" height="60" align="absmiddle"><img src="http://vitruvio.imss.fi.it/foto/mdl/V_2_ar.gif" alt="V.2.a" name="img0a" hspace="2" align="absmiddle" onclick="cambiaImg(0,'http://vitruvio.imss.fi.it/foto/mdl/V_2_ars.jpg', aggiustaApici('Leonardo da Vinci, Madrid Ms. I, 137r<br/>Five increasingly heavier weights (1, 2, 3, 4, 5) are applied centrally to equal rods. Leonardo is convinced that the curvature induced on the rods will increase proportionally.'), aggiustaApici('V.2.a'), '')" style="cursor:pointer; border-color:#98907d" onmouseover="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_ar.gif', true)" onmouseout="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_ar.gif', false)" border="1"><script>
				totIcone++;
				pathImm["img"+totIcone] = 'http://vitruvio.imss.fi.it/foto/mdl/V_2_ars.jpg';
				didascalie[totIcone]='Leonardo da Vinci, Madrid Ms. I, 137r<br/>Five increasingly heavier weights (1, 2, 3, 4, 5) are applied centrally to equal rods. Leonardo is convinced that the curvature induced on the rods will increase proportionally.';
				titolo[totIcone]='V.2.a';
				chiaveImm[totIcone]='13839';
				filmato[totIcone]='';
			</script></pre>
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<pre><img src="images/b.gif" alt="b" width="47" height="60" align="absmiddle"><img src="http://vitruvio.imss.fi.it/foto/mdl/V_2_b_apice1r.gif" alt="V.2.b1" name="img0b" hspace="2" align="absmiddle" onclick="cambiaImg(0,'http://vitruvio.imss.fi.it/foto/mdl/V_2_b_apice1rs.jpg', aggiustaApici('Leonardo da Vinci, Madrid Ms. I, 51r<br/>This drawing illustrates Leonardo|s conviction that the flexure of the cord of a cross-bow to which increasingly heavy weights are applied will not increase in the same proportion as the weights.'), aggiustaApici('V.2.b1'), '')" style="cursor:pointer; border-color:#98907d" onmouseover="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_b_apice1r.gif', true)" onmouseout="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_b_apice1r.gif', false)" border="1"><script>
				totIcone++;
				pathImm["img"+totIcone] = 'http://vitruvio.imss.fi.it/foto/mdl/V_2_b_apice1rs.jpg';
				didascalie[totIcone]='Leonardo da Vinci, Madrid Ms. I, 51r<br/>This drawing illustrates Leonardo|s conviction that the flexure of the cord of a cross-bow to which increasingly heavy weights are applied will not increase in the same proportion as the weights.';
				titolo[totIcone]='V.2.b1';
				chiaveImm[totIcone]='13841';
				filmato[totIcone]='';
			</script><img src="http://vitruvio.imss.fi.it/foto/mdl/V_2_b_apice2r.gif" alt="V.2.b2" name="img1b" hspace="2" align="absmiddle" onclick="cambiaImg(1,'http://vitruvio.imss.fi.it/foto/mdl/V_2_b_apice2rs.jpg', aggiustaApici('Two equal cross-bows under different loads<br/>Models after Leonardo da Vinci, Madrid Ms. I, 51r<br/>Opera Laboratori Fiorentini<br/>Leonardo states that inverse proportionality exists between the shooting power of the cross-bow and the angle formed by its stretched cords.'), aggiustaApici('V.2.b2'), '')" style="cursor:pointer; border-color:#98907d" onmouseover="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_b_apice2r.gif', true)" onmouseout="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_b_apice2r.gif', false)" border="1"><script>
				totIcone++;
				pathImm["img"+totIcone] = 'http://vitruvio.imss.fi.it/foto/mdl/V_2_b_apice2rs.jpg';
				didascalie[totIcone]='Two equal cross-bows under different loads<br/>Models after Leonardo da Vinci, Madrid Ms. I, 51r<br/>Opera Laboratori Fiorentini<br/>Leonardo states that inverse proportionality exists between the shooting power of the cross-bow and the angle formed by its stretched cords.';
				titolo[totIcone]='V.2.b2';
				chiaveImm[totIcone]='13842';
				filmato[totIcone]='';
			</script></pre>
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<pre><img src="images/c.gif" alt="c" width="47" height="60" align="absmiddle"><img src="http://vitruvio.imss.fi.it/foto/mdl/V_2_cr.gif" alt="V.2.c" name="img0c" hspace="2" align="absmiddle" onclick="cambiaImg(0,'http://vitruvio.imss.fi.it/foto/mdl/V_2_crs.jpg', aggiustaApici('Leonardo da Vinci, Codex Atlanticus, 311r<br/>The power of a shell fired by a mortar decreases proportionally with distance.'), aggiustaApici('V.2.c'), '')" style="cursor:pointer; border-color:#98907d" onmouseover="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_cr.gif', true)" onmouseout="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_cr.gif', false)" border="1"><script>
				totIcone++;
				pathImm["img"+totIcone] = 'http://vitruvio.imss.fi.it/foto/mdl/V_2_crs.jpg';
				didascalie[totIcone]='Leonardo da Vinci, Codex Atlanticus, 311r<br/>The power of a shell fired by a mortar decreases proportionally with distance.';
				titolo[totIcone]='V.2.c';
				chiaveImm[totIcone]='13843';
				filmato[totIcone]='';
			</script></pre>
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<li class="leoanteprime">
<pre><img src="images/d.gif" alt="d" width="47" height="60" align="absmiddle"><img src="http://vitruvio.imss.fi.it/foto/mdl/V_2_dr.gif" alt="V.2.d" name="img0d" hspace="2" align="absmiddle" onclick="cambiaImg(0,'http://vitruvio.imss.fi.it/foto/mdl/V_2_drs.jpg', aggiustaApici('Leonardo da Vinci, Madrid Ms. I, 85r<br/>The power of a spring is &quot;pyramidal&quot;, because it progressively decreases, as the spring unloads. To compensate for this loss (which alters the precision of spring watches), Leonardo designs a conical escapement, which reduces resistance proportionally.'), aggiustaApici('V.2.d'), '')" style="cursor:pointer; border-color:#98907d" onmouseover="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_dr.gif', true)" onmouseout="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_dr.gif', false)" border="1"><script>
				totIcone++;
				pathImm["img"+totIcone] = 'http://vitruvio.imss.fi.it/foto/mdl/V_2_drs.jpg';
				didascalie[totIcone]='Leonardo da Vinci, Madrid Ms. I, 85r<br/>The power of a spring is "pyramidal", because it progressively decreases, as the spring unloads. To compensate for this loss (which alters the precision of spring watches), Leonardo designs a conical escapement, which reduces resistance proportionally.';
				titolo[totIcone]='V.2.d';
				chiaveImm[totIcone]='13844';
				filmato[totIcone]='';
			</script></pre>
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<pre><img src="images/e.gif" alt="e" width="47" height="60" align="absmiddle"><img src="http://vitruvio.imss.fi.it/foto/mdl/V_2_er.gif" alt="V.2.e" name="img0e" hspace="2" align="absmiddle" onclick="cambiaImg(0,'http://vitruvio.imss.fi.it/foto/mdl/V_2_ers.jpg', aggiustaApici('Leonardo da Vinci, Madrid Ms. I, 147r<br/>&quot;Pyramidal&quot; diminution in the oscillations of a pendulum and in the rebounds of a ball.'), aggiustaApici('V.2.e'), '')" style="cursor:pointer; border-color:#98907d" onmouseover="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_er.gif', true)" onmouseout="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_er.gif', false)" border="1"><script>
				totIcone++;
				pathImm["img"+totIcone] = 'http://vitruvio.imss.fi.it/foto/mdl/V_2_ers.jpg';
				didascalie[totIcone]='Leonardo da Vinci, Madrid Ms. I, 147r<br/>"Pyramidal" diminution in the oscillations of a pendulum and in the rebounds of a ball.';
				titolo[totIcone]='V.2.e';
				chiaveImm[totIcone]='13845';
				filmato[totIcone]='';
			</script></pre>
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<pre><img src="images/f.gif" alt="f" width="47" height="60" align="absmiddle"><img src="http://vitruvio.imss.fi.it/foto/mdl/V_2_fr.gif" alt="V.2.f" name="img0f" hspace="2" align="absmiddle" onclick="cambiaImg(0,'http://vitruvio.imss.fi.it/foto/mdl/V_2_frs.jpg', aggiustaApici('Leonardo da Vinci, Ms. B, 90v<br/>&quot;Pyramidal&quot; diminution in sound: through a horn, a man produces a sound that rebounds from the stairway opposite the one on which he is seated. Just as would happen with a rebounding ball, the sound diminishes as it rebounds.'), aggiustaApici('V.2.f'), '')" style="cursor:pointer; border-color:#98907d" onmouseover="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_fr.gif', true)" onmouseout="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_fr.gif', false)" border="1"><script>
				totIcone++;
				pathImm["img"+totIcone] = 'http://vitruvio.imss.fi.it/foto/mdl/V_2_frs.jpg';
				didascalie[totIcone]='Leonardo da Vinci, Ms. B, 90v<br/>"Pyramidal" diminution in sound: through a horn, a man produces a sound that rebounds from the stairway opposite the one on which he is seated. Just as would happen with a rebounding ball, the sound diminishes as it rebounds.';
				titolo[totIcone]='V.2.f';
				chiaveImm[totIcone]='13846';
				filmato[totIcone]='';
			</script></pre>
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<li class="leoanteprime">
<pre><img src="images/g.gif" alt="g" width="47" height="60" align="absmiddle"><img src="http://vitruvio.imss.fi.it/foto/mdl/V_2_gr.gif" alt="V.2.g" name="img0g" hspace="2" align="absmiddle" onclick="cambiaImg(0,'http://vitruvio.imss.fi.it/foto/mdl/V_2_grs.jpg', aggiustaApici('Leonardo da Vinci, Ms. A, 20r<br/>&quot;Pyramidal&quot; diffusion of heat and cold: the Sun|s rays reflected from the focal point of a concave mirror multiply the heat. In the same way, cold increases in proportion to the number of bellows that blow air at the same point.'), aggiustaApici('V.2.g'), '')" style="cursor:pointer; border-color:#98907d" onmouseover="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_gr.gif', true)" onmouseout="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_gr.gif', false)" border="1"><script>
				totIcone++;
				pathImm["img"+totIcone] = 'http://vitruvio.imss.fi.it/foto/mdl/V_2_grs.jpg';
				didascalie[totIcone]='Leonardo da Vinci, Ms. A, 20r<br/>"Pyramidal" diffusion of heat and cold: the Sun|s rays reflected from the focal point of a concave mirror multiply the heat. In the same way, cold increases in proportion to the number of bellows that blow air at the same point.';
				titolo[totIcone]='V.2.g';
				chiaveImm[totIcone]='13847';
				filmato[totIcone]='';
			</script></pre>
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</div>
<div align="left">
<li class="leoanteprime">
<pre><img src="images/h.gif" alt="h" width="47" height="60" align="absmiddle"><img src="http://vitruvio.imss.fi.it/foto/mdl/V_2_hr.gif" alt="V.2.h" name="img0h" hspace="2" align="absmiddle" onclick="cambiaImg(0,'http://vitruvio.imss.fi.it/foto/mdl/V_2_hrs.jpg', aggiustaApici('Leonardo da Vinci, Ms. C, 22r<br/>Studies of relationships of proportionality between sources of light and the shadows generated by them.'), aggiustaApici('V.2.h'), '')" style="cursor:pointer; border-color:#98907d" onmouseover="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_hr.gif', true)" onmouseout="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_hr.gif', false)" border="1"><script>
				totIcone++;
				pathImm["img"+totIcone] = 'http://vitruvio.imss.fi.it/foto/mdl/V_2_hrs.jpg';
				didascalie[totIcone]='Leonardo da Vinci, Ms. C, 22r<br/>Studies of relationships of proportionality between sources of light and the shadows generated by them.';
				titolo[totIcone]='V.2.h';
				chiaveImm[totIcone]='13848';
				filmato[totIcone]='';
			</script></pre>
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<div align="left">
<li class="leoanteprime">
<pre><img src="images/i.gif" alt="i" width="47" height="60" align="absmiddle"><img src="http://vitruvio.imss.fi.it/foto/mdl/V_2_ir.gif" alt="V.2.i" name="img0i" hspace="2" align="absmiddle" onclick="cambiaImg(0,'http://vitruvio.imss.fi.it/foto/mdl/V_2_irs.jpg', aggiustaApici('Leonardo da Vinci, Madrid Ms. I, 134v<br/>&quot;Pyramidal&quot; relationship between hydraulic pressure and power of a jet of water.'), aggiustaApici('V.2.i'), '')" style="cursor:pointer; border-color:#98907d" onmouseover="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_ir.gif', true)" onmouseout="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_ir.gif', false)" border="1"><script>
				totIcone++;
				pathImm["img"+totIcone] = 'http://vitruvio.imss.fi.it/foto/mdl/V_2_irs.jpg';
				didascalie[totIcone]='Leonardo da Vinci, Madrid Ms. I, 134v<br/>"Pyramidal" relationship between hydraulic pressure and power of a jet of water.';
				titolo[totIcone]='V.2.i';
				chiaveImm[totIcone]='13849';
				filmato[totIcone]='';
			</script></pre>
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<li class="leoanteprime">
<pre><img src="images/j.gif" alt="j" width="47" height="60" align="absmiddle"><img src="http://vitruvio.imss.fi.it/foto/mdl/V_2_jr.gif" alt="V.2.j" name="img0j" hspace="2" align="absmiddle" onclick="cambiaImg(0,'http://vitruvio.imss.fi.it/foto/mdl/V_2_jrs.jpg', aggiustaApici('Leonardo da Vinci, Codex Atlanticus, 347r<br/>Diagram of the &quot;pyramidal&quot; diffusion of rays of light, odors, blows and magnets.'), aggiustaApici('V.2.j'), '')" style="cursor:pointer; border-color:#98907d" onmouseover="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_jr.gif', true)" onmouseout="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_jr.gif', false)" border="1"><script>
				totIcone++;
				pathImm["img"+totIcone] = 'http://vitruvio.imss.fi.it/foto/mdl/V_2_jrs.jpg';
				didascalie[totIcone]='Leonardo da Vinci, Codex Atlanticus, 347r<br/>Diagram of the "pyramidal" diffusion of rays of light, odors, blows and magnets.';
				titolo[totIcone]='V.2.j';
				chiaveImm[totIcone]='13850';
				filmato[totIcone]='';
			</script></pre>
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<div align="left">
<li class="leoanteprime">
<pre><img src="images/k.gif" alt="k" width="47" height="60" align="absmiddle"><img src="http://vitruvio.imss.fi.it/foto/mdl/V_2_k_apice2r.gif" alt="V.2.k2" name="img0k" hspace="2" align="absmiddle" onclick="cambiaImg(0,'http://vitruvio.imss.fi.it/foto/mdl/V_2_k_apice2rs.jpg', aggiustaApici('Cords of equal section with weights applied in proportion to length<br/>Model after Leonardo da Vinci, Forster Ms. II, 123v<br/>Opera Laboratori Fiorentini<br/>A cord of a certain length supports a weight of 200. One double this length supports a weight of 100. But if the latter is passed through a pulley, it supports a weight of 200.'), aggiustaApici('V.2.k2'), '')" style="cursor:pointer; border-color:#98907d" onmouseover="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_k_apice2r.gif', true)" onmouseout="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_k_apice2r.gif', false)" border="1"><script>
				totIcone++;
				pathImm["img"+totIcone] = 'http://vitruvio.imss.fi.it/foto/mdl/V_2_k_apice2rs.jpg';
				didascalie[totIcone]='Cords of equal section with weights applied in proportion to length<br/>Model after Leonardo da Vinci, Forster Ms. II, 123v<br/>Opera Laboratori Fiorentini<br/>A cord of a certain length supports a weight of 200. One double this length supports a weight of 100. But if the latter is passed through a pulley, it supports a weight of 200.';
				titolo[totIcone]='V.2.k2';
				chiaveImm[totIcone]='13852';
				filmato[totIcone]='';
			</script></pre>
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<div align="left">
<li class="leoanteprime">
<pre><img src="images/l.gif" alt="l" width="47" height="60" align="absmiddle"><img src="http://vitruvio.imss.fi.it/foto/mdl/V_2_l_apice1r.gif" alt="V.2.l1" name="img0l" hspace="2" align="absmiddle" onclick="cambiaImg(0,'http://vitruvio.imss.fi.it/foto/mdl/V_2_l_apice1rs.jpg', aggiustaApici('Francesco Gaffurio, <i>Theorica Musice</i><br/>Imaginary portrait of Pythagoras picking musical strings stretched to different degrees of tension.'), aggiustaApici('V.2.l1'), '')" style="cursor:pointer; border-color:#98907d" onmouseover="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_l_apice1r.gif', true)" onmouseout="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_l_apice1r.gif', false)" border="1"><script>
				totIcone++;
				pathImm["img"+totIcone] = 'http://vitruvio.imss.fi.it/foto/mdl/V_2_l_apice1rs.jpg';
				didascalie[totIcone]='Francesco Gaffurio, <i>Theorica Musice</i><br/>Imaginary portrait of Pythagoras picking musical strings stretched to different degrees of tension.';
				titolo[totIcone]='V.2.l1';
				chiaveImm[totIcone]='13853';
				filmato[totIcone]='';
			</script><img src="http://vitruvio.imss.fi.it/foto/mdl/V_2_l_apice2r.gif" alt="V.2.l2" name="img1l" hspace="2" align="absmiddle" onclick="cambiaImg(1,'http://vitruvio.imss.fi.it/foto/mdl/V_2_l_apice2rs.jpg', aggiustaApici('Monochord<br/>Model after Leonardo da Vinci, Forster Ms. II, 35v<br/>Opera Laboratori Fiorentini<br/>For Leonardo the tone produced by the vibration of the 4 strings of equal section and material varies in proportion to the tension to which they are subjected.'), aggiustaApici('V.2.l2'), '')" style="cursor:pointer; border-color:#98907d" onmouseover="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_l_apice2r.gif', true)" onmouseout="cambiaBordo('http://vitruvio.imss.fi.it/foto/mdl/V_2_l_apice2r.gif', false)" border="1"><script>
				totIcone++;
				pathImm["img"+totIcone] = 'http://vitruvio.imss.fi.it/foto/mdl/V_2_l_apice2rs.jpg';
				didascalie[totIcone]='Monochord<br/>Model after Leonardo da Vinci, Forster Ms. II, 35v<br/>Opera Laboratori Fiorentini<br/>For Leonardo the tone produced by the vibration of the 4 strings of equal section and material varies in proportion to the tension to which they are subjected.';
				titolo[totIcone]='V.2.l2';
				chiaveImm[totIcone]='13854';
				filmato[totIcone]='';
			</script></pre>
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