Harriot, Thomas, Mss. 6782

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          <pb file="add_6782_f267" o="267" n="533"/>
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            <p>
              <s xml:space="preserve">[
                <emph style="bf">Commentary:</emph>
              </s>
            </p>
            <p>
              <s xml:space="preserve"> In modern notation, binomes are numbers of the form
                <math>
                  <mstyle>
                    <msqrt>
                      <mrow>
                        <mi>m</mi>
                      </mrow>
                    </msqrt>
                    <mo>+</mo>
                    <msqrt>
                      <mrow>
                        <mi>n</mi>
                      </mrow>
                    </msqrt>
                  </mstyle>
                </math>
              where
                <math>
                  <mstyle>
                    <mi>m</mi>
                  </mstyle>
                </math>
              and
                <math>
                  <mstyle>
                    <mi>n</mi>
                  </mstyle>
                </math>
              are integers.
                <lb/>
                <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookX/bookX.html#defsII"> Book X; Definitions </ref>
              , Euclid defined six kinds of binomes, according to various relationships of
                <math>
                  <mstyle>
                    <mi>m</mi>
                  </mstyle>
                </math>
              to
                <math>
                  <mstyle>
                    <mi>n</mi>
                  </mstyle>
                </math>
              , which for Euclid were geometric lengths. In modern notation, the six binomes may be defined as follows.
                <lb/>
              Binome 1: a binome of the form
                <math>
                  <mstyle>
                    <mi>m</mi>
                    <mo>+</mo>
                    <msqrt>
                      <mrow>
                        <mi>n</mi>
                      </mrow>
                    </msqrt>
                  </mstyle>
                </math>
              with
                <math>
                  <mstyle>
                    <mi>m</mi>
                    <mo>></mo>
                    <msqrt>
                      <mrow>
                        <mi>n</mi>
                      </mrow>
                    </msqrt>
                  </mstyle>
                </math>
              , and
                <math>
                  <mstyle>
                    <mrow>
                      <msup>
                        <mi>m</mi>
                        <mn>2</mn>
                      </msup>
                    </mrow>
                    <mo>=</mo>
                    <mi>n</mi>
                    <mo>+</mo>
                    <mi>k</mi>
                  </mstyle>
                </math>
              where
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mi>m</mi>
                      </mrow>
                      <mrow>
                        <msqrt>
                          <mrow>
                            <mi>k</mi>
                          </mrow>
                        </msqrt>
                      </mrow>
                    </mfrac>
                  </mstyle>
                </math>
              is rational; for example
                <math>
                  <mstyle>
                    <mn>7</mn>
                    <mo>+</mo>
                    <msqrt>
                      <mrow>
                        <mn>4</mn>
                        <mn>8</mn>
                      </mrow>
                    </msqrt>
                  </mstyle>
                </math>
              .
                <lb/>
              Binome 2: a binome of the form
                <math>
                  <mstyle>
                    <msqrt>
                      <mrow>
                        <mi>m</mi>
                      </mrow>
                    </msqrt>
                    <mo>+</mo>
                    <mi>n</mi>
                  </mstyle>
                </math>
              with
                <math>
                  <mstyle>
                    <msqrt>
                      <mrow>
                        <mi>m</mi>
                      </mrow>
                    </msqrt>
                    <mo>></mo>
                    <mi>n</mi>
                  </mstyle>
                </math>
              , and
                <math>
                  <mstyle>
                    <mi>m</mi>
                    <mo>=</mo>
                    <mrow>
                      <msup>
                        <mi>n</mi>
                        <mn>2</mn>
                      </msup>
                    </mrow>
                    <mo>+</mo>
                    <mi>k</mi>
                  </mstyle>
                </math>
              where
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <msqrt>
                          <mrow>
                            <mi>m</mi>
                          </mrow>
                        </msqrt>
                      </mrow>
                      <mrow>
                        <msqrt>
                          <mrow>
                            <mi>k</mi>
                          </mrow>
                        </msqrt>
                      </mrow>
                    </mfrac>
                  </mstyle>
                </math>
              is rational; for example
                <math>
                  <mstyle>
                    <msqrt>
                      <mrow>
                        <mn>1</mn>
                        <mn>2</mn>
                      </mrow>
                    </msqrt>
                    <mo>+</mo>
                    <mn>3</mn>
                  </mstyle>
                </math>
              .
                <lb/>
              Binome 3: a binome of the form
                <math>
                  <mstyle>
                    <msqrt>
                      <mrow>
                        <mi>m</mi>
                      </mrow>
                    </msqrt>
                    <mo>+</mo>
                    <msqrt>
                      <mrow>
                        <mi>n</mi>
                      </mrow>
                    </msqrt>
                  </mstyle>
                </math>
              with
                <math>
                  <mstyle>
                    <msqrt>
                      <mrow>
                        <mi>m</mi>
                      </mrow>
                    </msqrt>
                    <mo>></mo>
                    <msqrt>
                      <mrow>
                        <mi>n</mi>
                      </mrow>
                    </msqrt>
                  </mstyle>
                </math>
              , and
                <math>
                  <mstyle>
                    <mi>m</mi>
                    <mo>=</mo>
                    <mi>n</mi>
                    <mo>+</mo>
                    <mi>k</mi>
                  </mstyle>
                </math>
              where
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <msqrt>
                          <mrow>
                            <mi>m</mi>
                          </mrow>
                        </msqrt>
                      </mrow>
                      <mrow>
                        <msqrt>
                          <mrow>
                            <mi>k</mi>
                          </mrow>
                        </msqrt>
                      </mrow>
                    </mfrac>
                  </mstyle>
                </math>
              is rational; for example
                <math>
                  <mstyle>
                    <msqrt>
                      <mrow>
                        <mn>8</mn>
                      </mrow>
                    </msqrt>
                    <mo>+</mo>
                    <msqrt>
                      <mrow>
                        <mn>6</mn>
                      </mrow>
                    </msqrt>
                  </mstyle>
                </math>
              .
                <lb/>
              Binome 4: a binome of the form
                <math>
                  <mstyle>
                    <mi>m</mi>
                    <mo>+</mo>
                    <msqrt>
                      <mrow>
                        <mi>n</mi>
                      </mrow>
                    </msqrt>
                  </mstyle>
                </math>
              with
                <math>
                  <mstyle>
                    <mi>m</mi>
                    <mo>></mo>
                    <msqrt>
                      <mrow>
                        <mi>n</mi>
                      </mrow>
                    </msqrt>
                  </mstyle>
                </math>
              , and
                <math>
                  <mstyle>
                    <mrow>
                      <msup>
                        <mi>m</mi>
                        <mn>2</mn>
                      </msup>
                    </mrow>
                    <mo>=</mo>
                    <mi>n</mi>
                    <mo>+</mo>
                    <mi>k</mi>
                  </mstyle>
                </math>
              where
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mi>m</mi>
                      </mrow>
                      <mrow>
                        <msqrt>
                          <mrow>
                            <mi>k</mi>
                          </mrow>
                        </msqrt>
                      </mrow>
                    </mfrac>
                  </mstyle>
                </math>
              is non-rational; for example
                <math>
                  <mstyle>
                    <mn>2</mn>
                    <mo>+</mo>
                    <msqrt>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </msqrt>
                  </mstyle>
                </math>
              .
                <lb/>
              Binome 5: a binome of the form
                <math>
                  <mstyle>
                    <msqrt>
                      <mrow>
                        <mi>m</mi>
                      </mrow>
                    </msqrt>
                    <mo>+</mo>
                    <mi>n</mi>
                  </mstyle>
                </math>
              with
                <math>
                  <mstyle>
                    <msqrt>
                      <mrow>
                        <mi>m</mi>
                      </mrow>
                    </msqrt>
                    <mo>></mo>
                    <mi>n</mi>
                  </mstyle>
                </math>
              , and
                <math>
                  <mstyle>
                    <mi>m</mi>
                    <mo>=</mo>
                    <mrow>
                      <msup>
                        <mi>n</mi>
                        <mn>2</mn>
                      </msup>
                    </mrow>
                    <mo>+</mo>
                    <mi>k</mi>
                  </mstyle>
                </math>
              where
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <msqrt>
                          <mrow>
                            <mi>m</mi>
                          </mrow>
                        </msqrt>
                      </mrow>
                      <mrow>
                        <msqrt>
                          <mrow>
                            <mi>k</mi>
                          </mrow>
                        </msqrt>
                      </mrow>
                    </mfrac>
                  </mstyle>
                </math>
              is non-rational; for example
                <math>
                  <mstyle>
                    <msqrt>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </msqrt>
                    <mo>+</mo>
                    <mn>1</mn>
                  </mstyle>
                </math>
              .
                <lb/>
              Binome 6: a binome of the form
                <math>
                  <mstyle>
                    <msqrt>
                      <mrow>
                        <mi>m</mi>
                      </mrow>
                    </msqrt>
                    <mo>+</mo>
                    <msqrt>
                      <mrow>
                        <mi>n</mi>
                      </mrow>
                    </msqrt>
                  </mstyle>
                </math>
              with
                <math>
                  <mstyle>
                    <msqrt>
                      <mrow>
                        <mi>m</mi>
                      </mrow>
                    </msqrt>
                    <mo>></mo>
                    <msqrt>
                      <mrow>
                        <mi>n</mi>
                      </mrow>
                    </msqrt>
                  </mstyle>
                </math>
              , and
                <math>
                  <mstyle>
                    <mi>m</mi>
                    <mo>=</mo>
                    <mi>n</mi>
                    <mo>+</mo>
                    <mi>k</mi>
                  </mstyle>
                </math>
              where
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <msqrt>
                          <mrow>
                            <mi>m</mi>
                          </mrow>
                        </msqrt>
                      </mrow>
                      <mrow>
                        <msqrt>
                          <mrow>
                            <mi>k</mi>
                          </mrow>
                        </msqrt>
                      </mrow>
                    </mfrac>
                  </mstyle>
                </math>
              is non-rational; for example
                <math>
                  <mstyle>
                    <msqrt>
                      <mrow>
                        <mn>3</mn>
                      </mrow>
                    </msqrt>
                    <mo>+</mo>
                    <msqrt>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </msqrt>
                  </mstyle>
                </math>
              .
                <lb/>
              Harriot made two further distinctions for binomes of the fifth and sixth kind according to
                <math>
                  <mstyle>
                    <mi>k</mi>
                  </mstyle>
                </math>
              is itself a square (type i) or not (type (ii).
                <lb/>
              In this and the following folio, Add MS
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/HSPGZ0AE&start=530&viewMode=image&pn=535"> f. </ref>
              , Harriot shows that the square of any binome is always a binome of the first kind. This folio shows his working for first, second, and third binomes. </s>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat"> Binomiorum quadrata, sunt binomia
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          Squares of binomes are binomes of the first ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> 1.
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            A binomial of the first ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"/>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"/>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> ut
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            as ]</s>
          </p>
        </div>
      </text>
    </echo>
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