Harriot, Thomas, Mss. 6785

List of thumbnails

< >
231
231 (116) 		232
232 (116v)
233
233 (117) 		234
234 (117v)
235
235 (118) 		236
236 (118v)
237
237 (119) 		238
238 (119v)
239
239 (120) 		240
240 (120v)
< >
page |< < (178v) of 882 > >|
    <echo version="1.0RC">
      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6785_f178v" o="178v" n="356"/>
          <div type="page_commentary" level="0" n="0">
            <p>
              <s xml:space="preserve">[
                <emph style="bf">Commentary:</emph>
              </s>
            </p>
            <p>
              <s xml:space="preserve"> Sums of some infinite progressions.
                <lb/>
              The first example
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>3</mn>
                      </mrow>
                      <mrow>
                        <mn>4</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>4</mn>
                      </mrow>
                      <mrow>
                        <mn>8</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>5</mn>
                      </mrow>
                      <mrow>
                        <mn>1</mn>
                        <mn>6</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                    <mo>=</mo>
                    <mn>4</mn>
                  </mstyle>
                </math>
              . From the similar examples shown on Add MS 6789
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/0VGM2B80/&start=80&viewMode=image&pn=87"> f. </ref>
              , we may assume that Harriot summed this as follows:
                <lb/>
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>4</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>8</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                    <mo>=</mo>
                    <mn>2</mn>
                  </mstyle>
                </math>
              ,
                <lb/>
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>4</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>8</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                    <mo>=</mo>
                    <mn>1</mn>
                  </mstyle>
                </math>
              ,
                <lb/>
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>4</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>8</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                    <mo>=</mo>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </mfrac>
                  </mstyle>
                </math>
              ,
                <lb/>
                <math>
                  <mstyle>
                    <mo>.</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                  </mstyle>
                </math>
              . The sums of these series form a geometric progression
                <math>
                  <mstyle>
                    <mn>2</mn>
                    <mo>+</mo>
                    <mn>1</mn>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>4</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                    <mo>=</mo>
                    <mn>4</mn>
                  </mstyle>
                </math>
              .
                <lb/>
              The second example is similar to the first.
                <lb/>
              The third example
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>3</mn>
                      </mrow>
                      <mrow>
                        <mn>3</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>6</mn>
                      </mrow>
                      <mrow>
                        <mn>9</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>2</mn>
                        <mn>7</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>2</mn>
                        <mn>0</mn>
                      </mrow>
                      <mrow>
                        <mn>8</mn>
                        <mn>1</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                    <mo>=</mo>
                    <mn>3</mn>
                    <mfrac>
                      <mrow>
                        <mn>3</mn>
                      </mrow>
                      <mrow>
                        <mn>4</mn>
                      </mrow>
                    </mfrac>
                  </mstyle>
                </math>
              . This can be rewritten as the sum of two separate series:
                <lb/>
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                      <mrow>
                        <mn>3</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>4</mn>
                      </mrow>
                      <mrow>
                        <mn>9</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>8</mn>
                      </mrow>
                      <mrow>
                        <mn>2</mn>
                        <mn>7</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                        <mn>6</mn>
                      </mrow>
                      <mrow>
                        <mn>8</mn>
                        <mn>1</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                  </mstyle>
                </math>
              . and
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>3</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                      <mrow>
                        <mn>9</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>3</mn>
                      </mrow>
                      <mrow>
                        <mn>2</mn>
                        <mn>7</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>4</mn>
                      </mrow>
                      <mrow>
                        <mn>8</mn>
                        <mn>1</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                  </mstyle>
                </math>
              . The first is a geometric progression whose sum is 3.
                <lb/>
              The second can be summed as in the first example, to give
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mn>3</mn>
                      </mrow>
                      <mrow>
                        <mn>4</mn>
                      </mrow>
                    </mfrac>
                  </mstyle>
                </math>
              .
                <lb/>
              Thus the total sum is
                <math>
                  <mstyle>
                    <mn>3</mn>
                    <mfrac>
                      <mrow>
                        <mn>3</mn>
                      </mrow>
                      <mrow>
                        <mn>4</mn>
                      </mrow>
                    </mfrac>
                  </mstyle>
                </math>
              .
                <lb/>
              The fourth example
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>5</mn>
                      </mrow>
                      <mrow>
                        <mn>6</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                        <mn>3</mn>
                      </mrow>
                      <mrow>
                        <mn>3</mn>
                        <mn>6</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>3</mn>
                        <mn>5</mn>
                      </mrow>
                      <mrow>
                        <mn>2</mn>
                        <mn>1</mn>
                        <mn>6</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                  </mstyle>
                </math>
              . This can be rewritten as the sum of two separate series:
                <lb/>
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                      <mrow>
                        <mn>6</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>4</mn>
                      </mrow>
                      <mrow>
                        <mn>3</mn>
                        <mn>6</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>9</mn>
                      </mrow>
                      <mrow>
                        <mn>2</mn>
                        <mn>1</mn>
                        <mn>6</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                  </mstyle>
                </math>
              . and
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>3</mn>
                      </mrow>
                      <mrow>
                        <mn>6</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>9</mn>
                      </mrow>
                      <mrow>
                        <mn>3</mn>
                        <mn>6</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>2</mn>
                        <mn>7</mn>
                      </mrow>
                      <mrow>
                        <mn>2</mn>
                        <mn>1</mn>
                        <mn>6</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                    <mo>.</mo>
                  </mstyle>
                </math>
              . The first is a geometric progression whose sum is 3.
                <lb/>
              The second is a geomteric progression whose sum is 2.
                <lb/>
              Thus the total sum is 5.
                <lb/>
              </s>
              <s xml:space="preserve">]</s>
            </p>
          </div>
        </div>
      </text>
    </echo>
Impressum