Alberti, Leone Battista, Architecture, 1755

Table of figures

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb xlink:href="003/01/276.jpg" pagenum="199"/>
            <p type="main">
              <s>THIS
                <emph type="italics"/>
              Quadruple
                <emph.end type="italics"/>
              may be alſo formed by
                <lb/>
              adding a
                <emph type="italics"/>
              Seſquialtera
                <emph.end type="italics"/>
              and a
                <emph type="italics"/>
              Seſquitertia
                <emph.end type="italics"/>
              to the
                <lb/>
                <emph type="italics"/>
              Duple;
                <emph.end type="italics"/>
              and how this is done, is manifeſt by
                <lb/>
              what we have ſaid above: But for its clearer
                <lb/>
              Explanation, we ſhall give a further Inſtance
                <lb/>
              of it here. </s>
              <s>The Number two, for Example,
                <lb/>
              by Means of a
                <emph type="italics"/>
              Seſquialtera
                <emph.end type="italics"/>
              is made three, which
                <lb/>
              by a
                <emph type="italics"/>
              Seſquitertia
                <emph.end type="italics"/>
              becomes four, which four
                <lb/>
              being doubled makes eight.
                <lb/>
                <arrow.to.target n="table14"/>
              </s>
            </p>
            <table>
              <table.target id="table14"/>
              <row>
                <cell/>
                <cell>00</cell>
                <cell/>
              </row>
              <row>
                <cell/>
                <cell>000</cell>
                <cell>
                  <emph type="italics"/>
                Seſquialtera
                  <emph.end type="italics"/>
                </cell>
              </row>
              <row>
                <cell>The
                  <emph type="italics"/>
                Quadruple.
                  <emph.end type="italics"/>
                </cell>
                <cell/>
                <cell/>
              </row>
              <row>
                <cell/>
                <cell>0000</cell>
                <cell>
                  <emph type="italics"/>
                Seſquitertia
                  <emph.end type="italics"/>
                </cell>
              </row>
              <row>
                <cell/>
                <cell>00000000</cell>
                <cell>doubled</cell>
              </row>
            </table>
            <p type="main">
              <s>OR rather in the following Manner. </s>
              <s>Let us
                <lb/>
              take the Number three; this being doubled
                <lb/>
              makes ſix, to which adding another three, we
                <lb/>
              have nine, and adding to this a third of itſelf,
                <lb/>
              it produces twelve, which anſwers to three in a
                <lb/>
                <emph type="italics"/>
              Quadruple
                <emph.end type="italics"/>
              Proportion.
                <lb/>
                <arrow.to.target n="table15"/>
              </s>
            </p>
            <table>
              <table.target id="table15"/>
              <row>
                <cell/>
                <cell>000</cell>
                <cell/>
              </row>
              <row>
                <cell/>
                <cell>000000</cell>
                <cell>doubled</cell>
              </row>
              <row>
                <cell>The
                  <emph type="italics"/>
                Quadruple
                  <emph.end type="italics"/>
                </cell>
                <cell/>
                <cell/>
              </row>
              <row>
                <cell/>
                <cell>000000000</cell>
                <cell>a third added</cell>
              </row>
              <row>
                <cell/>
                <cell>000000000000</cell>
                <cell>a third added</cell>
              </row>
            </table>
            <p type="main">
              <s>THE Architects make uſe of all the ſeveral
                <lb/>
              Proportions here ſet down, not confuſedly and
                <lb/>
              indiſtinctly, but in ſuch Manner as to be con­
                <lb/>
              ſtantly and every way agreeable to Harmony:
                <lb/>
              As, for Inſtance, in the Elevation of a Room
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              which is twice as long as broad, they make
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              uſe, not of thoſe Numbers which compoſe the
                <lb/>
              Triple, but of thoſe only which form the
                <lb/>
              Duple; and the ſame in a Room whoſe Length
                <lb/>
              is three Times its Breadth, employing only its
                <lb/>
              own proper Proportions, and no foreign ones,
                <lb/>
              that is to ſay, taking ſuch of the triple Pro­
                <lb/>
              greſſions above ſet down, as is moſt agreeable
                <lb/>
              to the Circumſtances of their Structure. </s>
              <s>There
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              are ſome other natural Proportions for the Uſe
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              of Structures, which are not borrowed from
                <lb/>
              Numbers, but from the Roots and Powers of
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              Squares. </s>
              <s>The Roots are the Sides of ſquare
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              Numbers: The Powers are the Areas of thoſe
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              Squares: The Multiplication of the Areas
                <lb/>
              produce the Cubes. </s>
              <s>The firſt of all Cubes,
                <lb/>
              whoſe Root is one, is conſecrated to the Deity,
                <lb/>
              becauſe, as it is derived from One, So it is
                <lb/>
              One every Way; to which we may add, that
                <lb/>
              it is the moſt ſtable and conſtant of all Fi­
                <lb/>
              gures, and the very Baſis of all the reſt. </s>
              <s>But
                <lb/>
              if, as ſome affirm, the Unite be no Number,
                <lb/>
              but only the Source of all others, we may then
                <lb/>
              ſuppoſe the firſt Number to be the Number
                <lb/>
              two. </s>
              <s>Taking this Number two for the Root,
                <lb/>
              the Areas will be four, which being raiſed up
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              to a Height equal to its Root, will produce a
                <lb/>
              Cube of eight; and from this Cube we may
                <lb/>
              gather the Rules for our Proportions; for here
                <lb/>
              in the firſt Place, we may conſider the Side of
                <lb/>
              the Cube, which is called the Cube Root,
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              whoſe Area will in Numbers be ſour, and the
                <lb/>
              compleat or entire Cube be as eight. </s>
              <s>In the
                <lb/>
              next Place we may conſider the Line drawn
                <lb/>
              from one Angle of the Cube to that which is
                <lb/>
              directly oppoſite to it, ſo as to divide the Area
                <lb/>
              of the Square into two equal Parts, and this is
                <lb/>
              called the Diagonal. </s>
              <s>What this amounts to
                <lb/>
              in Numbers is not known: Only it appears
                <lb/>
              to be the Root of an Area, which is as Eight
                <lb/>
              on every Side; beſides which it is the Diago­
                <lb/>
              nal of a Cube which is on every Side, as twelve,
                <lb/>
                <arrow.to.target n="marg48"/>
                <lb/>
                <emph type="italics"/>
              Fig.
                <emph.end type="italics"/>
              1.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg48"/>
              *</s>
            </p>
            <p type="main">
              <s>LASTLY, In a Triangle whoſe two ſhorteſt
                <lb/>
              Sides form a Right Angle, and one of them
                <lb/>
              the Root of an Area, which is every Way as
                <lb/>
              four, and the other of one, which is as twelve,
                <lb/>
              the longſt Side ſubtended oppoſite to that
                <lb/>
              Right Angle, will be the Root of an Area,
                <lb/>
              will be the Root of an Area, which is as ſix­
                <lb/>
                <arrow.to.target n="marg49"/>
                <lb/>
              teen
                <emph type="italics"/>
              Fig.
                <emph.end type="italics"/>
              2.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg49"/>
              *</s>
            </p>
            <p type="main">
              <s>THESE ſeveral Rules which we have here
                <lb/>
              ſet down for the determining of Proportions,
                <lb/>
              are the natural and proper Relations of Num­
                <lb/>
              bers and Quantities, and the general Method
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              for the Practice of them all is, that the ſhorteſt
                <lb/>
              Line be taken for the Breadth of the Area,
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              the longeſt for the Length, and the middle
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              Line for the Height, tho' ſometimes ſor the
                <lb/>
              Convenience of the Structure, they are inter­
                <lb/>
              changed. </s>
              <s>We are now to ſay ſomething of
                <lb/>
              the Rules of thoſe Proportions, which are not
                <lb/>
              derived from Harmony or the natural Pro­
                <lb/>
              portions of Bodies, but are borrowed elſewhere
                <lb/>
              for determining the three Relations of an
                <lb/>
              Apartment; and in order to this we are to
                <lb/>
              obſerve, that there are very uſeful Conſidera­
                <lb/>
              tions in Practice to be drawn from the Muſi­
                <lb/>
              cians, Geometers, and even the Arithmeticians,
                <lb/>
              of each of which we are now to ſpeak. </s>
              <s>Theſe
                <lb/>
              the Philoſophers call
                <emph type="italics"/>
              Mediocrates,
                <emph.end type="italics"/>
              or
                <emph type="italics"/>
              Means,
                <emph.end type="italics"/>
                <lb/>
              and the Rules for them are many and various;
                <lb/>
              but there are three particularly which are the
                <lb/>
              moſt eſteemed; of all which the Purpoſe is,
                <lb/>
              that the two Extreams being given, the middle
                <lb/>
              Mean or Number may correſpond with them
                <lb/>
              in a certain detemined Manner, or to uſe
                <lb/>
              ſuch an Expreſſion, with a regular Affinity.
                <lb/>
              </s>
              <s>Our Buſineſs, in this Enquiry, is to conſider
                <lb/>
              three Terms, whereof the two moſt remote
                <lb/>
              are one the greateſt, and the other the leaſt;
                <lb/>
              the third or mean Number muſt anſwer to
                <lb/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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