Alberti, Leone Battista, Architecture, 1755

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="003/01/274.jpg" pagenum="198"/>
              equal Length, all correſponding with one an­
                <lb/>
              other at Right Angles. </s>
              <s>The neareſt to this is
                <lb/>
              the
                <emph type="italics"/>
              Seſquialtera,
                <emph.end type="italics"/>
              and the
                <emph type="italics"/>
              Seſquitertian
                <emph.end type="italics"/>
              alſo may
                <lb/>
              be reckoned among the ſhorter Areas. </s>
              <s>Theſe
                <lb/>
              three Proportions therefore, which we may alſo
                <lb/>
              call ſimple, are proper for the ſmaller Plat­
                <lb/>
              forms. </s>
              <s>There are likewiſe three others, which
                <lb/>
              are proper for middling Platforms: The beſt
                <lb/>
              of all is the Double, and the next beſt is that
                <lb/>
              which is formed of the
                <emph type="italics"/>
              Seſquialtera
                <emph.end type="italics"/>
              doubled,
                <lb/>
              which is produced as follows: Having ſet
                <lb/>
              down the leaſt Number of the Area, as, for
                <lb/>
              Inſtance, four, lengthen it to the firſt
                <emph type="italics"/>
              Seſqui­
                <lb/>
              altera,
                <emph.end type="italics"/>
              which will make ſix, and then add the
                <lb/>
                <emph type="italics"/>
              Seſquialtera
                <emph.end type="italics"/>
              of this ſix, which will produce
                <lb/>
              nine. </s>
              <s>Thus the Length will exceed the Breadth
                <lb/>
              in a double Proportion, and one Tone more.
                <lb/>
                <arrow.to.target n="table7"/>
              </s>
            </p>
            <table>
              <table.target id="table7"/>
              <row>
                <cell>4 0000</cell>
                <cell/>
              </row>
              <row>
                <cell/>
                <cell>
                  <emph type="italics"/>
                Seſquialtera
                  <emph.end type="italics"/>
                </cell>
              </row>
              <row>
                <cell>6 000000</cell>
                <cell/>
              </row>
              <row>
                <cell>9 000000000</cell>
                <cell>
                  <emph type="italics"/>
                Seſquialtera
                  <emph.end type="italics"/>
                doubled</cell>
              </row>
            </table>
            <p type="main">
              <s>FOR moderate Platforms alſo, we may uſe
                <lb/>
              that Proportion which ariſes from the
                <emph type="italics"/>
              Seſqui­
                <lb/>
              tertian
                <emph.end type="italics"/>
              doubled in the ſame Manner as the for­
                <lb/>
              mer; wherein the Length and Breadth will
                <lb/>
              be as nine and ſixteen.
                <lb/>
                <arrow.to.target n="table8"/>
              </s>
            </p>
            <table>
              <table.target id="table8"/>
              <row>
                <cell>9 000000000</cell>
                <cell/>
              </row>
              <row>
                <cell/>
                <cell>
                  <emph type="italics"/>
                Seſquitertia
                  <emph.end type="italics"/>
                </cell>
              </row>
              <row>
                <cell>12 000000000000</cell>
                <cell/>
              </row>
              <row>
                <cell>16 0000000000000000</cell>
                <cell>
                  <emph type="italics"/>
                Seſquitertia
                  <emph.end type="italics"/>
                doubled</cell>
              </row>
            </table>
            <p type="main">
              <s>HERE the longer Line contains the ſhorter
                <lb/>
              twice, excluding one Tone of that ſhorter
                <lb/>
              Line. </s>
              <s>In the longeſt Areas we either add the
                <lb/>
                <emph type="italics"/>
              Duple
                <emph.end type="italics"/>
              to the
                <emph type="italics"/>
              Seſquialtera,
                <emph.end type="italics"/>
              which will produce
                <lb/>
              the
                <emph type="italics"/>
              Triple;
                <emph.end type="italics"/>
              or add the
                <emph type="italics"/>
              Seſquitertia
                <emph.end type="italics"/>
              to the
                <lb/>
                <emph type="italics"/>
              Duple,
                <emph.end type="italics"/>
              which will make the Proportion as three
                <lb/>
              to eight; or laſtly make the Lines correſpond
                <lb/>
              to each other in a
                <emph type="italics"/>
              Quadruple
                <emph.end type="italics"/>
              Proportion. </s>
              <s>We
                <lb/>
              have now ſpoke of the ſhorter Platforms,
                <lb/>
              wherein the Numbers anſwer to each other
                <lb/>
              equally, as two to three, or three to four, and
                <lb/>
              of the Middling, wherein they correſpond as
                <lb/>
              two to four, or as four to nine, or as nine to
                <lb/>
              ſixteen: And laſtly of the longeſt, wherein
                <lb/>
              the Numbers anſwer in a
                <emph type="italics"/>
              Triple
                <emph.end type="italics"/>
              or
                <emph type="italics"/>
              Quadruple
                <emph.end type="italics"/>
                <lb/>
              Proportion, or as three to eight. </s>
              <s>We may
                <lb/>
              join together or compound all the three Lines
                <lb/>
              of any Body whatſoever, by Means of theſe ſe­
                <lb/>
              veral Number, which are either innate with
                <lb/>
              Harmony itſelf, or produced from other
                <lb/>
              Proportions in a certain and regular Me­
                <lb/>
              thod. </s>
              <s>We find in Harmony thoſe Num­
                <lb/>
              bers from whoſe mutual Relations we may
                <lb/>
              form their ſeveral Proporions, as in the
                <emph type="italics"/>
              Duple,
                <emph.end type="italics"/>
                <lb/>
              the
                <emph type="italics"/>
              Triple
                <emph.end type="italics"/>
              and the
                <emph type="italics"/>
              Quadruple.
                <emph.end type="italics"/>
              </s>
              <s> For In­
                <lb/>
              ſtance, the
                <emph type="italics"/>
              Duple
                <emph.end type="italics"/>
              is formed of the ſimple
                <emph type="italics"/>
              Seſ­
                <lb/>
              quialtera,
                <emph.end type="italics"/>
              with the Addition of the
                <emph type="italics"/>
              Seſquitertia,
                <emph.end type="italics"/>
                <lb/>
              in the following Method. </s>
              <s>Let the leaſt Num­
                <lb/>
              ber of the
                <emph type="italics"/>
              Duple
                <emph.end type="italics"/>
              be two; the
                <emph type="italics"/>
              Seſquialtera
                <emph.end type="italics"/>
              of
                <lb/>
              this is three, and the
                <emph type="italics"/>
              Seſquitertia
                <emph.end type="italics"/>
              of this Num­
                <lb/>
              ber three is four, which is juſt the Double of
                <lb/>
              two before-mentioned.
                <lb/>
                <arrow.to.target n="table9"/>
              </s>
            </p>
            <table>
              <table.target id="table9"/>
              <row>
                <cell>00</cell>
                <cell/>
              </row>
              <row>
                <cell>000</cell>
                <cell>The
                  <emph type="italics"/>
                Seſquialtera
                  <emph.end type="italics"/>
                </cell>
              </row>
              <row>
                <cell>0000</cell>
                <cell>The
                  <emph type="italics"/>
                Seſquitertia
                  <emph.end type="italics"/>
                or
                  <emph type="italics"/>
                Duple
                  <emph.end type="italics"/>
                </cell>
              </row>
            </table>
            <p type="main">
              <s>OR elſe the ſame is done in the following
                <lb/>
              Manner: Let the ſmaller Number be, for In­
                <lb/>
              ſtance, three; I add one to make it a
                <emph type="italics"/>
              Seſqui­
                <lb/>
              tertia,
                <emph.end type="italics"/>
              and it becomes four, to which adding a
                <lb/>
                <emph type="italics"/>
              Seſquialtera,
                <emph.end type="italics"/>
              it makes it ſix, which, compared
                <lb/>
              to three, is juſt in a double Proportion.
                <lb/>
                <arrow.to.target n="table10"/>
              </s>
            </p>
            <table>
              <table.target id="table10"/>
              <row>
                <cell/>
                <cell>000</cell>
                <cell/>
              </row>
              <row>
                <cell>The
                  <emph type="italics"/>
                Duple
                  <emph.end type="italics"/>
                </cell>
                <cell>0000</cell>
                <cell>
                  <emph type="italics"/>
                Seſquitertia
                  <emph.end type="italics"/>
                </cell>
              </row>
              <row>
                <cell/>
                <cell>000000</cell>
                <cell>
                  <emph type="italics"/>
                Seſquialtera
                  <emph.end type="italics"/>
                </cell>
              </row>
            </table>
            <p type="main">
              <s>THE
                <emph type="italics"/>
              Triple
                <emph.end type="italics"/>
              is likewiſe made of the
                <emph type="italics"/>
              Duple,
                <emph.end type="italics"/>
                <lb/>
              and of the
                <emph type="italics"/>
              Seſquialtera
                <emph.end type="italics"/>
              joined together: For
                <lb/>
              Inſtance, let the ſmaller Number here be two;
                <lb/>
              this being doubled, makes four; to which
                <lb/>
              adding a
                <emph type="italics"/>
              Seſquialtera,
                <emph.end type="italics"/>
              it becomes ſix, which is
                <lb/>
              the
                <emph type="italics"/>
              Triple
                <emph.end type="italics"/>
              of two.
                <lb/>
                <arrow.to.target n="table11"/>
              </s>
            </p>
            <table>
              <table.target id="table11"/>
              <row>
                <cell/>
                <cell>00</cell>
                <cell/>
              </row>
              <row>
                <cell>The
                  <emph type="italics"/>
                Triple
                  <emph.end type="italics"/>
                </cell>
                <cell>0000</cell>
                <cell>doubled</cell>
              </row>
              <row>
                <cell/>
                <cell>000000</cell>
                <cell>
                  <emph type="italics"/>
                Seſquialtera
                  <emph.end type="italics"/>
                </cell>
              </row>
            </table>
            <p type="main">
              <s>OR the ſame Thing is done as follows;
                <lb/>
              placing the ſame Number of two for the
                <lb/>
              ſmaller Number, take the
                <emph type="italics"/>
              Seſquialtera,
                <emph.end type="italics"/>
              and
                <lb/>
              you will have three, which being doubled,
                <lb/>
              gives ſix, and ſo we ſhall have the
                <emph type="italics"/>
              Triple
                <emph.end type="italics"/>
              of
                <lb/>
              two.
                <lb/>
                <arrow.to.target n="table12"/>
              </s>
            </p>
            <table>
              <table.target id="table12"/>
              <row>
                <cell/>
                <cell>00</cell>
                <cell/>
              </row>
              <row>
                <cell>The
                  <emph type="italics"/>
                Triple
                  <emph.end type="italics"/>
                </cell>
                <cell>000</cell>
                <cell>
                  <emph type="italics"/>
                Seſquialtera
                  <emph.end type="italics"/>
                </cell>
              </row>
              <row>
                <cell/>
                <cell>000000</cell>
                <cell>doubled</cell>
              </row>
            </table>
            <p type="main">
              <s>BY Means of the ſame Extenſions we may
                <lb/>
              produce the
                <emph type="italics"/>
              Quadruple,
                <emph.end type="italics"/>
              by compounding one
                <lb/>
                <emph type="italics"/>
              Duple
                <emph.end type="italics"/>
              with another, ſince it is indeed nothing
                <lb/>
              more than the
                <emph type="italics"/>
              Duple
                <emph.end type="italics"/>
              doubled, which is alſo
                <lb/>
              called
                <emph type="italics"/>
              Diſdiapaſon,
                <emph.end type="italics"/>
              and is performed as follows:
                <lb/>
              Let the ſmaller Number here, for Inſtance, be
                <lb/>
              two; double this, and it makes the
                <emph type="italics"/>
              Diapaſon,
                <emph.end type="italics"/>
                <lb/>
              that is to ſay four, which is the
                <emph type="italics"/>
              Duple
                <emph.end type="italics"/>
              of two,
                <lb/>
              and doubling this four, it makes the
                <emph type="italics"/>
              Diſdiapa­
                <lb/>
              ſon,
                <emph.end type="italics"/>
              which is as eight to two.
                <lb/>
                <arrow.to.target n="table13"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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